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Theorem constr3trllem2 25945
Description: Lemma for constr3trl 25953. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
Hypotheses
Ref Expression
constr3cycl.f 𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}
constr3cycl.p 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})
Assertion
Ref Expression
constr3trllem2 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → Fun 𝐹)

Proof of Theorem constr3trllem2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 constr3cycl.f . . . . . 6 𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}
2 constr3cycl.p . . . . . 6 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})
31, 2constr3trllem1 25944 . . . . 5 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹 ∈ Word dom 𝐸)
4 wrdf 13111 . . . . 5 (𝐹 ∈ Word dom 𝐸𝐹:(0..^(#‘𝐹))⟶dom 𝐸)
53, 4syl 17 . . . 4 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹:(0..^(#‘𝐹))⟶dom 𝐸)
61, 2constr3lem2 25940 . . . . 5 (#‘𝐹) = 3
7 usgraf1o 25653 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1-onto→ran 𝐸)
8 3cycl3dv 25936 . . . . . . . . . . . . . 14 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐴𝐵𝐵𝐶𝐶𝐴))
9 usgraedgrnv 25672 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐴𝑉𝐵𝑉))
109ex 448 . . . . . . . . . . . . . . . . . . 19 (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝐴𝑉𝐵𝑉)))
11 usgraedgrnv 25672 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 USGrph 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝐶𝑉𝐴𝑉))
1211ex 448 . . . . . . . . . . . . . . . . . . 19 (𝑉 USGrph 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 → (𝐶𝑉𝐴𝑉)))
1310, 12anim12d 583 . . . . . . . . . . . . . . . . . 18 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉))))
1413com12 32 . . . . . . . . . . . . . . . . 17 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉))))
15143adant2 1072 . . . . . . . . . . . . . . . 16 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉))))
1615impcom 444 . . . . . . . . . . . . . . 15 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)))
171, 2constr3lem5 25942 . . . . . . . . . . . . . . . . 17 ((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴}))
1817jctl 561 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))
1918exp31 627 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))))
2016, 19mpancom 699 . . . . . . . . . . . . . 14 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))))
218, 20mpd 15 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))))
2221ex 448 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))))
237, 22mpid 42 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))))
2423imp 443 . . . . . . . . . 10 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))
2524adantl 480 . . . . . . . . 9 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))
26 c0ex 9890 . . . . . . . . . . 11 0 ∈ V
27 1ex 9891 . . . . . . . . . . 11 1 ∈ V
28 2z 11242 . . . . . . . . . . 11 2 ∈ ℤ
2926, 27, 283pm3.2i 1231 . . . . . . . . . 10 (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ)
30 eqidd 2610 . . . . . . . . . . . . . . 15 ((𝐹‘0) = (𝐹‘0) → 0 = 0)
3130a1i 11 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘0) = (𝐹‘0) → 0 = 0))
32 f1of1 6034 . . . . . . . . . . . . . . . . . . 19 (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1→ran 𝐸)
3332adantl 480 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → 𝐸:dom 𝐸1-1→ran 𝐸)
34 3simpa 1050 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
3534adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
3635ad3antlr 762 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
37 f1ocnvfvrneq 6419 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶}))
3833, 36, 37syl2anc 690 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶}))
39 simpl 471 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → (𝐴𝑉𝐵𝑉))
40 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝑉𝐵𝑉) → 𝐵𝑉)
41 simpl 471 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐶𝑉𝐴𝑉) → 𝐶𝑉)
4240, 41anim12i 587 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → (𝐵𝑉𝐶𝑉))
43 preq12bg 4321 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
4439, 42, 43syl2anc 690 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
45 df-ne 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
46 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐴 = 𝐵 → (𝐴 = 𝐵 → 0 = 1))
4745, 46sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴𝐵 → (𝐴 = 𝐵 → 0 = 1))
48473ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 0 = 1))
4948com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
5049adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 = 𝐵𝐵 = 𝐶) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
51 df-ne 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐶𝐴 ↔ ¬ 𝐶 = 𝐴)
52 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐶 = 𝐴 → (𝐶 = 𝐴 → 0 = 1))
5351, 52sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐶𝐴 → (𝐶 = 𝐴 → 0 = 1))
54533ad2ant3 1076 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐶 = 𝐴 → 0 = 1))
5554com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐶 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
5655eqcoms 2617 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐶 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
5756adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 = 𝐶𝐵 = 𝐵) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
5850, 57jaoi 392 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
5944, 58syl6bi 241 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1)))
6059com23 83 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 0 = 1)))
6160adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 0 = 1)))
6261imp 443 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 0 = 1))
6362adantr 479 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 0 = 1))
6438, 63syld 45 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → 0 = 1))
6564adantl 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → 0 = 1))
66 eqeq12 2622 . . . . . . . . . . . . . . . . . 18 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶})) → ((𝐹‘0) = (𝐹‘1) ↔ (𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶})))
67663adant3 1073 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘0) = (𝐹‘1) ↔ (𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶})))
6867imbi1d 329 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘0) = (𝐹‘1) → 0 = 1) ↔ ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → 0 = 1)))
6968adantr 479 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘0) = (𝐹‘1) → 0 = 1) ↔ ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → 0 = 1)))
7065, 69mpbird 245 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘0) = (𝐹‘1) → 0 = 1))
71 3simpb 1051 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
7271adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
7372ad3antlr 762 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
74 f1ocnvfvrneq 6419 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → {𝐴, 𝐵} = {𝐶, 𝐴}))
7533, 73, 74syl2anc 690 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → {𝐴, 𝐵} = {𝐶, 𝐴}))
76 preq12bg 4321 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐴, 𝐵} = {𝐶, 𝐴} ↔ ((𝐴 = 𝐶𝐵 = 𝐴) ∨ (𝐴 = 𝐴𝐵 = 𝐶))))
77 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐴 = 𝐵 → (𝐴 = 𝐵 → 0 = 2))
7845, 77sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴𝐵 → (𝐴 = 𝐵 → 0 = 2))
79783ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 0 = 2))
8079com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
8180eqcoms 2617 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
8281adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 = 𝐶𝐵 = 𝐴) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
83 df-ne 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐵𝐶 ↔ ¬ 𝐵 = 𝐶)
84 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐵 = 𝐶 → (𝐵 = 𝐶 → 0 = 2))
8583, 84sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐵𝐶 → (𝐵 = 𝐶 → 0 = 2))
86853ad2ant2 1075 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐵 = 𝐶 → 0 = 2))
8786com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 = 𝐶 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
8887adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 = 𝐴𝐵 = 𝐶) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
8982, 88jaoi 392 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 = 𝐶𝐵 = 𝐴) ∨ (𝐴 = 𝐴𝐵 = 𝐶)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
9076, 89syl6bi 241 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐴, 𝐵} = {𝐶, 𝐴} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2)))
9190com23 83 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐴, 𝐵} = {𝐶, 𝐴} → 0 = 2)))
9291adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐴, 𝐵} = {𝐶, 𝐴} → 0 = 2)))
9392imp 443 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐴, 𝐵} = {𝐶, 𝐴} → 0 = 2))
9493adantr 479 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐴, 𝐵} = {𝐶, 𝐴} → 0 = 2))
9575, 94syld 45 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → 0 = 2))
9695adantl 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → 0 = 2))
97 eqeq12 2622 . . . . . . . . . . . . . . . . . 18 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘0) = (𝐹‘2) ↔ (𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴})))
98973adant2 1072 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘0) = (𝐹‘2) ↔ (𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴})))
9998imbi1d 329 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘0) = (𝐹‘2) → 0 = 2) ↔ ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → 0 = 2)))
10099adantr 479 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘0) = (𝐹‘2) → 0 = 2) ↔ ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → 0 = 2)))
10196, 100mpbird 245 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘0) = (𝐹‘2) → 0 = 2))
10231, 70, 1013jca 1234 . . . . . . . . . . . . 13 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘0) = (𝐹‘0) → 0 = 0) ∧ ((𝐹‘0) = (𝐹‘1) → 0 = 1) ∧ ((𝐹‘0) = (𝐹‘2) → 0 = 2)))
103102adantl 480 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (((𝐹‘0) = (𝐹‘0) → 0 = 0) ∧ ((𝐹‘0) = (𝐹‘1) → 0 = 1) ∧ ((𝐹‘0) = (𝐹‘2) → 0 = 2)))
104 fveq2 6088 . . . . . . . . . . . . . . . 16 (𝑦 = 0 → (𝐹𝑦) = (𝐹‘0))
105104eqeq2d 2619 . . . . . . . . . . . . . . 15 (𝑦 = 0 → ((𝐹‘0) = (𝐹𝑦) ↔ (𝐹‘0) = (𝐹‘0)))
106 eqeq2 2620 . . . . . . . . . . . . . . 15 (𝑦 = 0 → (0 = 𝑦 ↔ 0 = 0))
107105, 106imbi12d 332 . . . . . . . . . . . . . 14 (𝑦 = 0 → (((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ ((𝐹‘0) = (𝐹‘0) → 0 = 0)))
108 fveq2 6088 . . . . . . . . . . . . . . . 16 (𝑦 = 1 → (𝐹𝑦) = (𝐹‘1))
109108eqeq2d 2619 . . . . . . . . . . . . . . 15 (𝑦 = 1 → ((𝐹‘0) = (𝐹𝑦) ↔ (𝐹‘0) = (𝐹‘1)))
110 eqeq2 2620 . . . . . . . . . . . . . . 15 (𝑦 = 1 → (0 = 𝑦 ↔ 0 = 1))
111109, 110imbi12d 332 . . . . . . . . . . . . . 14 (𝑦 = 1 → (((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ ((𝐹‘0) = (𝐹‘1) → 0 = 1)))
112 fveq2 6088 . . . . . . . . . . . . . . . 16 (𝑦 = 2 → (𝐹𝑦) = (𝐹‘2))
113112eqeq2d 2619 . . . . . . . . . . . . . . 15 (𝑦 = 2 → ((𝐹‘0) = (𝐹𝑦) ↔ (𝐹‘0) = (𝐹‘2)))
114 eqeq2 2620 . . . . . . . . . . . . . . 15 (𝑦 = 2 → (0 = 𝑦 ↔ 0 = 2))
115113, 114imbi12d 332 . . . . . . . . . . . . . 14 (𝑦 = 2 → (((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ ((𝐹‘0) = (𝐹‘2) → 0 = 2)))
116107, 111, 115raltpg 4182 . . . . . . . . . . . . 13 ((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ (((𝐹‘0) = (𝐹‘0) → 0 = 0) ∧ ((𝐹‘0) = (𝐹‘1) → 0 = 1) ∧ ((𝐹‘0) = (𝐹‘2) → 0 = 2))))
117116adantr 479 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ (((𝐹‘0) = (𝐹‘0) → 0 = 0) ∧ ((𝐹‘0) = (𝐹‘1) → 0 = 1) ∧ ((𝐹‘0) = (𝐹‘2) → 0 = 2))))
118103, 117mpbird 245 . . . . . . . . . . 11 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → ∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦))
119 pm3.22 463 . . . . . . . . . . . . . . . . . . . . 21 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
1201193adant3 1073 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
121120adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
122121ad3antlr 762 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
123 f1ocnvfvrneq 6419 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸)) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → {𝐵, 𝐶} = {𝐴, 𝐵}))
12433, 122, 123syl2anc 690 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → {𝐵, 𝐶} = {𝐴, 𝐵}))
125 preq12bg 4321 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵𝑉𝐶𝑉) ∧ (𝐴𝑉𝐵𝑉)) → ({𝐵, 𝐶} = {𝐴, 𝐵} ↔ ((𝐵 = 𝐴𝐶 = 𝐵) ∨ (𝐵 = 𝐵𝐶 = 𝐴))))
12642, 39, 125syl2anc 690 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐵, 𝐶} = {𝐴, 𝐵} ↔ ((𝐵 = 𝐴𝐶 = 𝐵) ∨ (𝐵 = 𝐵𝐶 = 𝐴))))
127 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐴 = 𝐵 → (𝐴 = 𝐵 → 1 = 0))
12845, 127sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴𝐵 → (𝐴 = 𝐵 → 1 = 0))
1291283ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 1 = 0))
130129com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
131130eqcoms 2617 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
132131adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 = 𝐴𝐶 = 𝐵) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
133 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐶 = 𝐴 → (𝐶 = 𝐴 → 1 = 0))
13451, 133sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐶𝐴 → (𝐶 = 𝐴 → 1 = 0))
1351343ad2ant3 1076 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐶 = 𝐴 → 1 = 0))
136135com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐶 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
137136adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 = 𝐵𝐶 = 𝐴) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
138132, 137jaoi 392 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵 = 𝐴𝐶 = 𝐵) ∨ (𝐵 = 𝐵𝐶 = 𝐴)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
139126, 138syl6bi 241 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐵, 𝐶} = {𝐴, 𝐵} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0)))
140139com23 83 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐵, 𝐶} = {𝐴, 𝐵} → 1 = 0)))
141140adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐵, 𝐶} = {𝐴, 𝐵} → 1 = 0)))
142141imp 443 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐵, 𝐶} = {𝐴, 𝐵} → 1 = 0))
143142adantr 479 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐵, 𝐶} = {𝐴, 𝐵} → 1 = 0))
144124, 143syld 45 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → 1 = 0))
145144adantl 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → 1 = 0))
146 eqeq12 2622 . . . . . . . . . . . . . . . . . . 19 (((𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘0) = (𝐸‘{𝐴, 𝐵})) → ((𝐹‘1) = (𝐹‘0) ↔ (𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵})))
147146ancoms 467 . . . . . . . . . . . . . . . . . 18 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶})) → ((𝐹‘1) = (𝐹‘0) ↔ (𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵})))
1481473adant3 1073 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘1) = (𝐹‘0) ↔ (𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵})))
149148imbi1d 329 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘1) = (𝐹‘0) → 1 = 0) ↔ ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → 1 = 0)))
150149adantr 479 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘1) = (𝐹‘0) → 1 = 0) ↔ ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → 1 = 0)))
151145, 150mpbird 245 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘1) = (𝐹‘0) → 1 = 0))
152 eqidd 2610 . . . . . . . . . . . . . . 15 ((𝐹‘1) = (𝐹‘1) → 1 = 1)
153152a1i 11 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘1) = (𝐹‘1) → 1 = 1))
154 3simpc 1052 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
155154adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
156155ad3antlr 762 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
157 f1ocnvfvrneq 6419 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → {𝐵, 𝐶} = {𝐶, 𝐴}))
15833, 156, 157syl2anc 690 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → {𝐵, 𝐶} = {𝐶, 𝐴}))
159 preq12bg 4321 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵𝑉𝐶𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐵, 𝐶} = {𝐶, 𝐴} ↔ ((𝐵 = 𝐶𝐶 = 𝐴) ∨ (𝐵 = 𝐴𝐶 = 𝐶))))
16042, 159sylancom 697 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐵, 𝐶} = {𝐶, 𝐴} ↔ ((𝐵 = 𝐶𝐶 = 𝐴) ∨ (𝐵 = 𝐴𝐶 = 𝐶))))
161 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐵 = 𝐶 → (𝐵 = 𝐶 → 1 = 2))
16283, 161sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐵𝐶 → (𝐵 = 𝐶 → 1 = 2))
1631623ad2ant2 1075 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐵 = 𝐶 → 1 = 2))
164163com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 = 𝐶 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
165164adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 = 𝐶𝐶 = 𝐴) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
166 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐴 = 𝐵 → (𝐴 = 𝐵 → 1 = 2))
16745, 166sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴𝐵 → (𝐴 = 𝐵 → 1 = 2))
1681673ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 1 = 2))
169168com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
170169eqcoms 2617 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
171170adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 = 𝐴𝐶 = 𝐶) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
172165, 171jaoi 392 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵 = 𝐶𝐶 = 𝐴) ∨ (𝐵 = 𝐴𝐶 = 𝐶)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
173160, 172syl6bi 241 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐵, 𝐶} = {𝐶, 𝐴} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2)))
174173com23 83 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐵, 𝐶} = {𝐶, 𝐴} → 1 = 2)))
175174adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐵, 𝐶} = {𝐶, 𝐴} → 1 = 2)))
176175imp 443 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐵, 𝐶} = {𝐶, 𝐴} → 1 = 2))
177176adantr 479 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐵, 𝐶} = {𝐶, 𝐴} → 1 = 2))
178158, 177syld 45 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → 1 = 2))
179178adantl 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → 1 = 2))
180 eqeq12 2622 . . . . . . . . . . . . . . . . . 18 (((𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘1) = (𝐹‘2) ↔ (𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴})))
1811803adant1 1071 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘1) = (𝐹‘2) ↔ (𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴})))
182181imbi1d 329 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘1) = (𝐹‘2) → 1 = 2) ↔ ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → 1 = 2)))
183182adantr 479 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘1) = (𝐹‘2) → 1 = 2) ↔ ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → 1 = 2)))
184179, 183mpbird 245 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘1) = (𝐹‘2) → 1 = 2))
185151, 153, 1843jca 1234 . . . . . . . . . . . . 13 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘1) = (𝐹‘0) → 1 = 0) ∧ ((𝐹‘1) = (𝐹‘1) → 1 = 1) ∧ ((𝐹‘1) = (𝐹‘2) → 1 = 2)))
186185adantl 480 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (((𝐹‘1) = (𝐹‘0) → 1 = 0) ∧ ((𝐹‘1) = (𝐹‘1) → 1 = 1) ∧ ((𝐹‘1) = (𝐹‘2) → 1 = 2)))
187104eqeq2d 2619 . . . . . . . . . . . . . . 15 (𝑦 = 0 → ((𝐹‘1) = (𝐹𝑦) ↔ (𝐹‘1) = (𝐹‘0)))
188 eqeq2 2620 . . . . . . . . . . . . . . 15 (𝑦 = 0 → (1 = 𝑦 ↔ 1 = 0))
189187, 188imbi12d 332 . . . . . . . . . . . . . 14 (𝑦 = 0 → (((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ ((𝐹‘1) = (𝐹‘0) → 1 = 0)))
190108eqeq2d 2619 . . . . . . . . . . . . . . 15 (𝑦 = 1 → ((𝐹‘1) = (𝐹𝑦) ↔ (𝐹‘1) = (𝐹‘1)))
191 eqeq2 2620 . . . . . . . . . . . . . . 15 (𝑦 = 1 → (1 = 𝑦 ↔ 1 = 1))
192190, 191imbi12d 332 . . . . . . . . . . . . . 14 (𝑦 = 1 → (((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ ((𝐹‘1) = (𝐹‘1) → 1 = 1)))
193112eqeq2d 2619 . . . . . . . . . . . . . . 15 (𝑦 = 2 → ((𝐹‘1) = (𝐹𝑦) ↔ (𝐹‘1) = (𝐹‘2)))
194 eqeq2 2620 . . . . . . . . . . . . . . 15 (𝑦 = 2 → (1 = 𝑦 ↔ 1 = 2))
195193, 194imbi12d 332 . . . . . . . . . . . . . 14 (𝑦 = 2 → (((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ ((𝐹‘1) = (𝐹‘2) → 1 = 2)))
196189, 192, 195raltpg 4182 . . . . . . . . . . . . 13 ((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ (((𝐹‘1) = (𝐹‘0) → 1 = 0) ∧ ((𝐹‘1) = (𝐹‘1) → 1 = 1) ∧ ((𝐹‘1) = (𝐹‘2) → 1 = 2))))
197196adantr 479 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ (((𝐹‘1) = (𝐹‘0) → 1 = 0) ∧ ((𝐹‘1) = (𝐹‘1) → 1 = 1) ∧ ((𝐹‘1) = (𝐹‘2) → 1 = 2))))
198186, 197mpbird 245 . . . . . . . . . . 11 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦))
199 pm3.22 463 . . . . . . . . . . . . . . . . . . . . 21 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
2001993adant2 1072 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
201200adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
202201ad3antlr 762 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
203 f1ocnvfvrneq 6419 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸)) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → {𝐶, 𝐴} = {𝐴, 𝐵}))
20433, 202, 203syl2anc 690 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → {𝐶, 𝐴} = {𝐴, 𝐵}))
205 preq12bg 4321 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐶𝑉𝐴𝑉) ∧ (𝐴𝑉𝐵𝑉)) → ({𝐶, 𝐴} = {𝐴, 𝐵} ↔ ((𝐶 = 𝐴𝐴 = 𝐵) ∨ (𝐶 = 𝐵𝐴 = 𝐴))))
206205ancoms 467 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐶, 𝐴} = {𝐴, 𝐵} ↔ ((𝐶 = 𝐴𝐴 = 𝐵) ∨ (𝐶 = 𝐵𝐴 = 𝐴))))
207 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐴 = 𝐵 → (𝐴 = 𝐵 → 2 = 0))
20845, 207sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴𝐵 → (𝐴 = 𝐵 → 2 = 0))
2092083ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 2 = 0))
210209com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
211210adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 = 𝐴𝐴 = 𝐵) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
212 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐵 = 𝐶 → (𝐵 = 𝐶 → 2 = 0))
21383, 212sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐵𝐶 → (𝐵 = 𝐶 → 2 = 0))
2142133ad2ant2 1075 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐵 = 𝐶 → 2 = 0))
215214com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 = 𝐶 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
216215eqcoms 2617 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐶 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
217216adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 = 𝐵𝐴 = 𝐴) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
218211, 217jaoi 392 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 = 𝐴𝐴 = 𝐵) ∨ (𝐶 = 𝐵𝐴 = 𝐴)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
219206, 218syl6bi 241 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐶, 𝐴} = {𝐴, 𝐵} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0)))
220219com23 83 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐶, 𝐴} = {𝐴, 𝐵} → 2 = 0)))
221220adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐶, 𝐴} = {𝐴, 𝐵} → 2 = 0)))
222221imp 443 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐶, 𝐴} = {𝐴, 𝐵} → 2 = 0))
223222adantr 479 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐶, 𝐴} = {𝐴, 𝐵} → 2 = 0))
224204, 223syld 45 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → 2 = 0))
225224adantl 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → 2 = 0))
226 eqeq12 2622 . . . . . . . . . . . . . . . . . . 19 (((𝐹‘2) = (𝐸‘{𝐶, 𝐴}) ∧ (𝐹‘0) = (𝐸‘{𝐴, 𝐵})) → ((𝐹‘2) = (𝐹‘0) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵})))
227226ancoms 467 . . . . . . . . . . . . . . . . . 18 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘2) = (𝐹‘0) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵})))
2282273adant2 1072 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘2) = (𝐹‘0) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵})))
229228imbi1d 329 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘2) = (𝐹‘0) → 2 = 0) ↔ ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → 2 = 0)))
230229adantr 479 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘2) = (𝐹‘0) → 2 = 0) ↔ ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → 2 = 0)))
231225, 230mpbird 245 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘2) = (𝐹‘0) → 2 = 0))
232 pm3.22 463 . . . . . . . . . . . . . . . . . . . . 21 (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
2332323adant1 1071 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
234233adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
235234ad3antlr 762 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
236 f1ocnvfvrneq 6419 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → {𝐶, 𝐴} = {𝐵, 𝐶}))
23733, 235, 236syl2anc 690 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → {𝐶, 𝐴} = {𝐵, 𝐶}))
238 simpr 475 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → (𝐶𝑉𝐴𝑉))
239 preq12bg 4321 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐶𝑉𝐴𝑉) ∧ (𝐵𝑉𝐶𝑉)) → ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ ((𝐶 = 𝐵𝐴 = 𝐶) ∨ (𝐶 = 𝐶𝐴 = 𝐵))))
240238, 42, 239syl2anc 690 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ ((𝐶 = 𝐵𝐴 = 𝐶) ∨ (𝐶 = 𝐶𝐴 = 𝐵))))
241 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐶 = 𝐴 → (𝐶 = 𝐴 → 2 = 1))
24251, 241sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐶𝐴 → (𝐶 = 𝐴 → 2 = 1))
2432423ad2ant3 1076 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐶 = 𝐴 → 2 = 1))
244243com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐶 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
245244eqcoms 2617 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐶 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
246245adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 = 𝐵𝐴 = 𝐶) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
247 pm2.21 118 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐴 = 𝐵 → (𝐴 = 𝐵 → 2 = 1))
24845, 247sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴𝐵 → (𝐴 = 𝐵 → 2 = 1))
2492483ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 2 = 1))
250249com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
251250adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 = 𝐶𝐴 = 𝐵) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
252246, 251jaoi 392 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 = 𝐵𝐴 = 𝐶) ∨ (𝐶 = 𝐶𝐴 = 𝐵)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
253240, 252syl6bi 241 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐶, 𝐴} = {𝐵, 𝐶} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1)))
254253com23 83 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐶, 𝐴} = {𝐵, 𝐶} → 2 = 1)))
255254adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐶, 𝐴} = {𝐵, 𝐶} → 2 = 1)))
256255imp 443 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐶, 𝐴} = {𝐵, 𝐶} → 2 = 1))
257256adantr 479 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐶, 𝐴} = {𝐵, 𝐶} → 2 = 1))
258237, 257syld 45 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → 2 = 1))
259258adantl 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → 2 = 1))
260 eqeq12 2622 . . . . . . . . . . . . . . . . . . 19 (((𝐹‘2) = (𝐸‘{𝐶, 𝐴}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶})) → ((𝐹‘2) = (𝐹‘1) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶})))
261260ancoms 467 . . . . . . . . . . . . . . . . . 18 (((𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘2) = (𝐹‘1) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶})))
2622613adant1 1071 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘2) = (𝐹‘1) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶})))
263262imbi1d 329 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘2) = (𝐹‘1) → 2 = 1) ↔ ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → 2 = 1)))
264263adantr 479 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘2) = (𝐹‘1) → 2 = 1) ↔ ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → 2 = 1)))
265259, 264mpbird 245 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘2) = (𝐹‘1) → 2 = 1))
266 eqidd 2610 . . . . . . . . . . . . . . 15 ((𝐹‘2) = (𝐹‘2) → 2 = 2)
267266a1i 11 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘2) = (𝐹‘2) → 2 = 2))
268231, 265, 2673jca 1234 . . . . . . . . . . . . 13 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘2) = (𝐹‘0) → 2 = 0) ∧ ((𝐹‘2) = (𝐹‘1) → 2 = 1) ∧ ((𝐹‘2) = (𝐹‘2) → 2 = 2)))
269268adantl 480 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (((𝐹‘2) = (𝐹‘0) → 2 = 0) ∧ ((𝐹‘2) = (𝐹‘1) → 2 = 1) ∧ ((𝐹‘2) = (𝐹‘2) → 2 = 2)))
270104eqeq2d 2619 . . . . . . . . . . . . . . 15 (𝑦 = 0 → ((𝐹‘2) = (𝐹𝑦) ↔ (𝐹‘2) = (𝐹‘0)))
271 eqeq2 2620 . . . . . . . . . . . . . . 15 (𝑦 = 0 → (2 = 𝑦 ↔ 2 = 0))
272270, 271imbi12d 332 . . . . . . . . . . . . . 14 (𝑦 = 0 → (((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦) ↔ ((𝐹‘2) = (𝐹‘0) → 2 = 0)))
273108eqeq2d 2619 . . . . . . . . . . . . . . 15 (𝑦 = 1 → ((𝐹‘2) = (𝐹𝑦) ↔ (𝐹‘2) = (𝐹‘1)))
274 eqeq2 2620 . . . . . . . . . . . . . . 15 (𝑦 = 1 → (2 = 𝑦 ↔ 2 = 1))
275273, 274imbi12d 332 . . . . . . . . . . . . . 14 (𝑦 = 1 → (((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦) ↔ ((𝐹‘2) = (𝐹‘1) → 2 = 1)))
276112eqeq2d 2619 . . . . . . . . . . . . . . 15 (𝑦 = 2 → ((𝐹‘2) = (𝐹𝑦) ↔ (𝐹‘2) = (𝐹‘2)))
277 eqeq2 2620 . . . . . . . . . . . . . . 15 (𝑦 = 2 → (2 = 𝑦 ↔ 2 = 2))
278276, 277imbi12d 332 . . . . . . . . . . . . . 14 (𝑦 = 2 → (((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦) ↔ ((𝐹‘2) = (𝐹‘2) → 2 = 2)))
279272, 275, 278raltpg 4182 . . . . . . . . . . . . 13 ((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦) ↔ (((𝐹‘2) = (𝐹‘0) → 2 = 0) ∧ ((𝐹‘2) = (𝐹‘1) → 2 = 1) ∧ ((𝐹‘2) = (𝐹‘2) → 2 = 2))))
280279adantr 479 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦) ↔ (((𝐹‘2) = (𝐹‘0) → 2 = 0) ∧ ((𝐹‘2) = (𝐹‘1) → 2 = 1) ∧ ((𝐹‘2) = (𝐹‘2) → 2 = 2))))
281269, 280mpbird 245 . . . . . . . . . . 11 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦))
282118, 198, 2813jca 1234 . . . . . . . . . 10 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
28329, 282mpan 701 . . . . . . . . 9 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
28425, 283syl 17 . . . . . . . 8 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
285 fveq2 6088 . . . . . . . . . . . . 13 (𝑥 = 0 → (𝐹𝑥) = (𝐹‘0))
286285eqeq1d 2611 . . . . . . . . . . . 12 (𝑥 = 0 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹‘0) = (𝐹𝑦)))
287 eqeq1 2613 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
288286, 287imbi12d 332 . . . . . . . . . . 11 (𝑥 = 0 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦)))
289288ralbidv 2968 . . . . . . . . . 10 (𝑥 = 0 → (∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦)))
290 fveq2 6088 . . . . . . . . . . . . 13 (𝑥 = 1 → (𝐹𝑥) = (𝐹‘1))
291290eqeq1d 2611 . . . . . . . . . . . 12 (𝑥 = 1 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹‘1) = (𝐹𝑦)))
292 eqeq1 2613 . . . . . . . . . . . 12 (𝑥 = 1 → (𝑥 = 𝑦 ↔ 1 = 𝑦))
293291, 292imbi12d 332 . . . . . . . . . . 11 (𝑥 = 1 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦)))
294293ralbidv 2968 . . . . . . . . . 10 (𝑥 = 1 → (∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦)))
295 fveq2 6088 . . . . . . . . . . . . 13 (𝑥 = 2 → (𝐹𝑥) = (𝐹‘2))
296295eqeq1d 2611 . . . . . . . . . . . 12 (𝑥 = 2 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹‘2) = (𝐹𝑦)))
297 eqeq1 2613 . . . . . . . . . . . 12 (𝑥 = 2 → (𝑥 = 𝑦 ↔ 2 = 𝑦))
298296, 297imbi12d 332 . . . . . . . . . . 11 (𝑥 = 2 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
299298ralbidv 2968 . . . . . . . . . 10 (𝑥 = 2 → (∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
300289, 294, 299raltpg 4182 . . . . . . . . 9 ((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) → (∀𝑥 ∈ {0, 1, 2}∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦))))
30129, 300ax-mp 5 . . . . . . . 8 (∀𝑥 ∈ {0, 1, 2}∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
302284, 301sylibr 222 . . . . . . 7 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ∀𝑥 ∈ {0, 1, 2}∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
303 fzo0to3tp 12376 . . . . . . . . 9 (0..^3) = {0, 1, 2}
304303a1i 11 . . . . . . . 8 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (0..^3) = {0, 1, 2})
305304raleqdv 3120 . . . . . . . 8 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
306304, 305raleqbidv 3128 . . . . . . 7 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (∀𝑥 ∈ (0..^3)∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ {0, 1, 2}∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
307302, 306mpbird 245 . . . . . 6 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ∀𝑥 ∈ (0..^3)∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
308 oveq2 6535 . . . . . . . 8 ((#‘𝐹) = 3 → (0..^(#‘𝐹)) = (0..^3))
309308raleqdv 3120 . . . . . . . 8 ((#‘𝐹) = 3 → (∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
310308, 309raleqbidv 3128 . . . . . . 7 ((#‘𝐹) = 3 → (∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ (0..^3)∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
311310adantr 479 . . . . . 6 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ (0..^3)∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
312307, 311mpbird 245 . . . . 5 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3136, 312mpan 701 . . . 4 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
314 dff13 6394 . . . 4 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3155, 313, 314sylanbrc 694 . . 3 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸)
316 df-f1 5795 . . 3 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun 𝐹))
317315, 316sylib 206 . 2 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun 𝐹))
318317simprd 477 1 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  Vcvv 3172  cun 3537  {cpr 4126  {ctp 4128  cop 4130   class class class wbr 4577  ccnv 5027  dom cdm 5028  ran crn 5029  Fun wfun 5784  wf 5786  1-1wf1 5787  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  0cc0 9792  1c1 9793  2c2 10917  3c3 10918  cz 11210  ..^cfzo 12289  #chash 12934  Word cword 13092   USGrph cusg 25625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-usgra 25628
This theorem is referenced by:  constr3trl  25953
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