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Theorem isomuspgrlem1 44211
Description: Lemma 1 for isomuspgr 44218. (Contributed by AV, 29-Nov-2022.)
Hypotheses
Ref Expression
isomushgr.v 𝑉 = (Vtx‘𝐴)
isomushgr.w 𝑊 = (Vtx‘𝐵)
isomushgr.e 𝐸 = (Edg‘𝐴)
isomushgr.k 𝐾 = (Edg‘𝐵)
Assertion
Ref Expression
isomuspgrlem1 (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) → ({(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾 → {𝑎, 𝑏} ∈ 𝐸))
Distinct variable groups:   𝐴,𝑒,𝑓,𝑔   𝐵,𝑒,𝑓,𝑔   𝑒,𝐸,𝑔   𝑔,𝐾   𝑒,𝑉,𝑔   𝑒,𝑊,𝑔   𝑎,𝑏,𝑔,𝑓
Allowed substitution hints:   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐸(𝑓,𝑎,𝑏)   𝐾(𝑒,𝑓,𝑎,𝑏)   𝑉(𝑓,𝑎,𝑏)   𝑊(𝑓,𝑎,𝑏)

Proof of Theorem isomuspgrlem1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 486 . . . . 5 ((𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒)) → 𝑔:𝐸1-1-onto𝐾)
21ad2antlr 726 . . . 4 (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) → 𝑔:𝐸1-1-onto𝐾)
3 f1ocnvdm 7031 . . . 4 ((𝑔:𝐸1-1-onto𝐾 ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸)
42, 3sylan 583 . . 3 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸)
5 uspgrupgr 26967 . . . . . . 7 (𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph)
65adantr 484 . . . . . 6 ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → 𝐴 ∈ UPGraph)
76ad4antr 731 . . . . 5 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → 𝐴 ∈ UPGraph)
8 isomushgr.v . . . . . 6 𝑉 = (Vtx‘𝐴)
9 isomushgr.e . . . . . 6 𝐸 = (Edg‘𝐴)
108, 9upgredg 26928 . . . . 5 ((𝐴 ∈ UPGraph ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) → ∃𝑥𝑉𝑦𝑉 (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦})
117, 10sylan 583 . . . 4 (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) → ∃𝑥𝑉𝑦𝑉 (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦})
12 eleq1 2903 . . . . . . . . . . 11 ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸 ↔ {𝑥, 𝑦} ∈ 𝐸))
1312biimpd 232 . . . . . . . . . 10 ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸 → {𝑥, 𝑦} ∈ 𝐸))
1413com12 32 . . . . . . . . 9 ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸 → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → {𝑥, 𝑦} ∈ 𝐸))
1514ad2antlr 726 . . . . . . . 8 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → {𝑥, 𝑦} ∈ 𝐸))
1615imp 410 . . . . . . 7 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦}) → {𝑥, 𝑦} ∈ 𝐸)
171ad6antlr 736 . . . . . . . . . . 11 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑔:𝐸1-1-onto𝐾)
18 simpr 488 . . . . . . . . . . 11 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → {𝑥, 𝑦} ∈ 𝐸)
19 simpr 488 . . . . . . . . . . . 12 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾)
2019ad5ant12 755 . . . . . . . . . . 11 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾)
2117, 18, 203jca 1125 . . . . . . . . . 10 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑔:𝐸1-1-onto𝐾 ∧ {𝑥, 𝑦} ∈ 𝐸 ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾))
22 f1ocnvfvb 7026 . . . . . . . . . 10 ((𝑔:𝐸1-1-onto𝐾 ∧ {𝑥, 𝑦} ∈ 𝐸 ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦}))
2321, 22syl 17 . . . . . . . . 9 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦}))
24 imaeq2 5913 . . . . . . . . . . . . . . . 16 (𝑒 = {𝑥, 𝑦} → (𝑓𝑒) = (𝑓 “ {𝑥, 𝑦}))
25 fveq2 6659 . . . . . . . . . . . . . . . 16 (𝑒 = {𝑥, 𝑦} → (𝑔𝑒) = (𝑔‘{𝑥, 𝑦}))
2624, 25eqeq12d 2840 . . . . . . . . . . . . . . 15 (𝑒 = {𝑥, 𝑦} → ((𝑓𝑒) = (𝑔𝑒) ↔ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
2726rspccv 3606 . . . . . . . . . . . . . 14 (∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
2827adantl 485 . . . . . . . . . . . . 13 ((𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒)) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
2928adantl 485 . . . . . . . . . . . 12 ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
3029ad4antr 731 . . . . . . . . . . 11 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})))
3130imp 410 . . . . . . . . . 10 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦}))
32 eqeq1 2828 . . . . . . . . . . . . 13 ((𝑔‘{𝑥, 𝑦}) = (𝑓 “ {𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)}))
3332eqcoms 2832 . . . . . . . . . . . 12 ((𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦}) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)}))
3433adantl 485 . . . . . . . . . . 11 ((((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) ∧ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)}))
35 f1ofn 6605 . . . . . . . . . . . . . . . . . 18 (𝑓:𝑉1-1-onto𝑊𝑓 Fn 𝑉)
3635ad6antlr 736 . . . . . . . . . . . . . . . . 17 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → 𝑓 Fn 𝑉)
37 simpl 486 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑦𝑉) → 𝑥𝑉)
3837adantl 485 . . . . . . . . . . . . . . . . 17 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑉)
39 simpr 488 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑦𝑉) → 𝑦𝑉)
4039adantl 485 . . . . . . . . . . . . . . . . 17 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑉)
4136, 38, 403jca 1125 . . . . . . . . . . . . . . . 16 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → (𝑓 Fn 𝑉𝑥𝑉𝑦𝑉))
4241adantr 484 . . . . . . . . . . . . . . 15 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑓 Fn 𝑉𝑥𝑉𝑦𝑉))
43 fnimapr 6736 . . . . . . . . . . . . . . 15 ((𝑓 Fn 𝑉𝑥𝑉𝑦𝑉) → (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑥), (𝑓𝑦)})
4442, 43syl 17 . . . . . . . . . . . . . 14 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑥), (𝑓𝑦)})
4544eqeq1d 2826 . . . . . . . . . . . . 13 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} ↔ {(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)}))
46 fvex 6672 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑥) ∈ V
47 fvex 6672 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑦) ∈ V
48 fvex 6672 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑎) ∈ V
49 fvex 6672 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓𝑏) ∈ V
5046, 47, 48, 49preq12b 4766 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} ↔ (((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∨ ((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎))))
51 f1of1 6603 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓:𝑉1-1-onto𝑊𝑓:𝑉1-1𝑊)
52 simpl 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎𝑉𝑏𝑉) → 𝑎𝑉)
5352, 37anim12ci 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑥𝑉𝑎𝑉))
54 f1veqaeq 7005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑥𝑉𝑎𝑉)) → ((𝑓𝑥) = (𝑓𝑎) → 𝑥 = 𝑎))
5551, 53, 54syl2anr 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑥) = (𝑓𝑎) → 𝑥 = 𝑎))
56 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎𝑉𝑏𝑉) → 𝑏𝑉)
5756, 39anim12ci 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑦𝑉𝑏𝑉))
58 f1veqaeq 7005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑦𝑉𝑏𝑉)) → ((𝑓𝑦) = (𝑓𝑏) → 𝑦 = 𝑏))
5951, 57, 58syl2anr 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑦) = (𝑓𝑏) → 𝑦 = 𝑏))
6055, 59anim12d 611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → (((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) → (𝑥 = 𝑎𝑦 = 𝑏)))
6160impcom 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → (𝑥 = 𝑎𝑦 = 𝑏))
6261orcd 870 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)))
6362ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎))))
6456, 37anim12ci 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑥𝑉𝑏𝑉))
65 f1veqaeq 7005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑥𝑉𝑏𝑉)) → ((𝑓𝑥) = (𝑓𝑏) → 𝑥 = 𝑏))
6651, 64, 65syl2anr 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑥) = (𝑓𝑏) → 𝑥 = 𝑏))
6752, 39anim12ci 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑦𝑉𝑎𝑉))
68 f1veqaeq 7005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓:𝑉1-1𝑊 ∧ (𝑦𝑉𝑎𝑉)) → ((𝑓𝑦) = (𝑓𝑎) → 𝑦 = 𝑎))
6951, 67, 68syl2anr 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑓𝑦) = (𝑓𝑎) → 𝑦 = 𝑎))
7066, 69anim12d 611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → (((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) → (𝑥 = 𝑏𝑦 = 𝑎)))
7170impcom 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → (𝑥 = 𝑏𝑦 = 𝑎))
7271olcd 871 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) ∧ (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊)) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)))
7372ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎)) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎))))
7463, 73jaoi 854 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∨ ((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎))) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎))))
75 vex 3483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑥 ∈ V
76 vex 3483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ∈ V
77 vex 3483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑎 ∈ V
78 vex 3483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑏 ∈ V
7975, 76, 77, 78preq12b 4766 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ ((𝑥 = 𝑎𝑦 = 𝑏) ∨ (𝑥 = 𝑏𝑦 = 𝑎)))
8074, 79syl6ibr 255 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑓𝑥) = (𝑓𝑎) ∧ (𝑓𝑦) = (𝑓𝑏)) ∨ ((𝑓𝑥) = (𝑓𝑏) ∧ (𝑓𝑦) = (𝑓𝑎))) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → {𝑥, 𝑦} = {𝑎, 𝑏}))
8150, 80sylbi 220 . . . . . . . . . . . . . . . . . . . . . . . 24 ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → {𝑥, 𝑦} = {𝑎, 𝑏}))
8281com12 32 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑓:𝑉1-1-onto𝑊) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏}))
8382expcom 417 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓:𝑉1-1-onto𝑊 → (((𝑎𝑉𝑏𝑉) ∧ (𝑥𝑉𝑦𝑉)) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
8483expd 419 . . . . . . . . . . . . . . . . . . . . 21 (𝑓:𝑉1-1-onto𝑊 → ((𝑎𝑉𝑏𝑉) → ((𝑥𝑉𝑦𝑉) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏}))))
8584ad2antlr 726 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) → ((𝑎𝑉𝑏𝑉) → ((𝑥𝑉𝑦𝑉) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏}))))
8685imp 410 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) → ((𝑥𝑉𝑦𝑉) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
8786adantr 484 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → ((𝑥𝑉𝑦𝑉) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
8887adantr 484 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) → ((𝑥𝑉𝑦𝑉) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑥, 𝑦} = {𝑎, 𝑏})))
8988imp31 421 . . . . . . . . . . . . . . . 16 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)}) → {𝑥, 𝑦} = {𝑎, 𝑏})
9089eleq1d 2900 . . . . . . . . . . . . . . 15 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)}) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
9190biimpd 232 . . . . . . . . . . . . . 14 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)}) → ({𝑥, 𝑦} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸))
9291impancom 455 . . . . . . . . . . . . 13 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ({(𝑓𝑥), (𝑓𝑦)} = {(𝑓𝑎), (𝑓𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
9345, 92sylbid 243 . . . . . . . . . . . 12 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
9493adantr 484 . . . . . . . . . . 11 ((((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) ∧ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})) → ((𝑓 “ {𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
9534, 94sylbid 243 . . . . . . . . . 10 ((((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) ∧ (𝑓 “ {𝑥, 𝑦}) = (𝑔‘{𝑥, 𝑦})) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
9631, 95mpdan 686 . . . . . . . . 9 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑔‘{𝑥, 𝑦}) = {(𝑓𝑎), (𝑓𝑏)} → {𝑎, 𝑏} ∈ 𝐸))
9723, 96sylbird 263 . . . . . . . 8 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → {𝑎, 𝑏} ∈ 𝐸))
9897impancom 455 . . . . . . 7 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦}) → ({𝑥, 𝑦} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸))
9916, 98mpd 15 . . . . . 6 (((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦}) → {𝑎, 𝑏} ∈ 𝐸)
10099ex 416 . . . . 5 ((((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) ∧ (𝑥𝑉𝑦𝑉)) → ((𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → {𝑎, 𝑏} ∈ 𝐸))
101100rexlimdvva 3287 . . . 4 (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) → (∃𝑥𝑉𝑦𝑉 (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) = {𝑥, 𝑦} → {𝑎, 𝑏} ∈ 𝐸))
10211, 101mpd 15 . . 3 (((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) ∧ (𝑔‘{(𝑓𝑎), (𝑓𝑏)}) ∈ 𝐸) → {𝑎, 𝑏} ∈ 𝐸)
1034, 102mpdan 686 . 2 ((((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) ∧ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → {𝑎, 𝑏} ∈ 𝐸)
104103ex 416 1 (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) → ({(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾 → {𝑎, 𝑏} ∈ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wrex 3134  {cpr 4552  ccnv 5542  cima 5546   Fn wfn 6339  1-1wf1 6341  1-1-ontowf1o 6343  cfv 6344  Vtxcvtx 26787  Edgcedg 26838  UPGraphcupgr 26871  USPGraphcuspgr 26939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7104  df-ov 7149  df-oprab 7150  df-mpo 7151  df-om 7572  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9323  df-card 9361  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11695  df-n0 11893  df-xnn0 11963  df-z 11977  df-uz 12239  df-fz 12893  df-hash 13694  df-edg 26839  df-upgr 26873  df-uspgr 26941
This theorem is referenced by:  isomuspgr  44218
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