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Theorem brfi1indALT 13080
Description: Alternate proof of brfi1ind 13079, which does not use brfi1uzind 13078. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
brfi1ind.r Rel 𝐺
brfi1ind.f 𝐹 ∈ V
brfi1ind.1 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
brfi1ind.2 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
brfi1ind.3 ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
brfi1ind.4 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
brfi1ind.base ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)
brfi1ind.step ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
Assertion
Ref Expression
brfi1indALT ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)
Distinct variable groups:   𝑒,𝐸,𝑛,𝑣   𝑓,𝐹,𝑤   𝑒,𝐺,𝑓,𝑛,𝑣,𝑤,𝑦   𝑒,𝑉,𝑛,𝑣   𝜓,𝑓,𝑛,𝑤,𝑦   𝜃,𝑒,𝑛,𝑣   𝜒,𝑓,𝑤   𝜑,𝑒,𝑛,𝑣
Allowed substitution hints:   𝜑(𝑦,𝑤,𝑓)   𝜓(𝑣,𝑒)   𝜒(𝑦,𝑣,𝑒,𝑛)   𝜃(𝑦,𝑤,𝑓)   𝐸(𝑦,𝑤,𝑓)   𝐹(𝑦,𝑣,𝑒,𝑛)   𝑉(𝑦,𝑤,𝑓)

Proof of Theorem brfi1indALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hashcl 12958 . . 3 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
2 df-clel 2602 . . . 4 ((#‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0))
3 eqeq2 2617 . . . . . . . . . . . . . 14 (𝑥 = 0 → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = 0))
43anbi2d 735 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = 0)))
54imbi1d 329 . . . . . . . . . . . 12 (𝑥 = 0 → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)))
652albidv 1837 . . . . . . . . . . 11 (𝑥 = 0 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)))
7 eqeq2 2617 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = 𝑦))
87anbi2d 735 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦)))
98imbi1d 329 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓)))
1092albidv 1837 . . . . . . . . . . 11 (𝑥 = 𝑦 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓)))
11 eqeq2 2617 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 + 1) → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = (𝑦 + 1)))
1211anbi2d 735 . . . . . . . . . . . . 13 (𝑥 = (𝑦 + 1) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1))))
1312imbi1d 329 . . . . . . . . . . . 12 (𝑥 = (𝑦 + 1) → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
14132albidv 1837 . . . . . . . . . . 11 (𝑥 = (𝑦 + 1) → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
15 eqeq2 2617 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = 𝑛))
1615anbi2d 735 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛)))
1716imbi1d 329 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓)))
18172albidv 1837 . . . . . . . . . . 11 (𝑥 = 𝑛 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓)))
19 brfi1ind.base . . . . . . . . . . . 12 ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)
2019gen2 1713 . . . . . . . . . . 11 𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)
21 breq12 4579 . . . . . . . . . . . . . . 15 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑣𝐺𝑒𝑤𝐺𝑓))
22 fveq2 6085 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
2322eqeq1d 2608 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑤 → ((#‘𝑣) = 𝑦 ↔ (#‘𝑤) = 𝑦))
2423adantr 479 . . . . . . . . . . . . . . 15 ((𝑣 = 𝑤𝑒 = 𝑓) → ((#‘𝑣) = 𝑦 ↔ (#‘𝑤) = 𝑦))
2521, 24anbi12d 742 . . . . . . . . . . . . . 14 ((𝑣 = 𝑤𝑒 = 𝑓) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) ↔ (𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦)))
26 brfi1ind.2 . . . . . . . . . . . . . 14 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
2725, 26imbi12d 332 . . . . . . . . . . . . 13 ((𝑣 = 𝑤𝑒 = 𝑓) → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓) ↔ ((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃)))
2827cbval2v 2268 . . . . . . . . . . . 12 (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓) ↔ ∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃))
29 nn0re 11145 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0𝑦 ∈ ℝ)
30 1re 9892 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℝ
3130a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → 1 ∈ ℝ)
32 nn0ge0 11162 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
33 0lt1 10396 . . . . . . . . . . . . . . . . . . . . 21 0 < 1
3433a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → 0 < 1)
3529, 31, 32, 34addgegt0d 10447 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
3635adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
37 simpr 475 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → (#‘𝑣) = (𝑦 + 1))
3836, 37breqtrrd 4602 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (#‘𝑣))
3938adantrl 747 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1))) → 0 < (#‘𝑣))
40 vex 3172 . . . . . . . . . . . . . . . . . . 19 𝑣 ∈ V
41 hashgt0elex 12999 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ∃𝑛 𝑛𝑣)
42 brfi1ind.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
4340a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑣 ∈ V)
44 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑛𝑣)
45 simpl 471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑦 ∈ ℕ0)
46 brfi1indlem 13076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0) → ((#‘𝑣) = (𝑦 + 1) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
4743, 44, 45, 46syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑦 ∈ ℕ0𝑛𝑣) → ((#‘𝑣) = (𝑦 + 1) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
4847imp 443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (#‘(𝑣 ∖ {𝑛})) = 𝑦)
49 peano2nn0 11177 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
5049ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑦 + 1) ∈ ℕ0)
5150ad2antlr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑦 + 1) ∈ ℕ0)
52 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑣𝐺𝑒)
53 simplrr 796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (#‘𝑣) = (𝑦 + 1))
54 simprlr 798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) → 𝑛𝑣)
5554adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑛𝑣)
5652, 53, 553jca 1234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
5751, 56jca 552 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → ((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)))
58 difexg 4727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
5940, 58ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑣 ∖ {𝑛}) ∈ V
60 brfi1ind.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 𝐹 ∈ V
61 breq12 4579 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑤𝐺𝑓 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
62 fveq2 6085 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
6362eqeq1d 2608 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑤 = (𝑣 ∖ {𝑛}) → ((#‘𝑤) = 𝑦 ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
6463adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((#‘𝑤) = 𝑦 ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
6561, 64anbi12d 742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) ↔ ((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦)))
66 brfi1ind.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
6765, 66imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ↔ (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
6867spc2gv 3265 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ V) → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
6959, 60, 68mp2an 703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
7069expdimp 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜒))
7170ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜒))
72 brfi1ind.step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
7357, 71, 72syl6an 565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜓))
7473exp41 635 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑣𝐺𝑒 → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜓)))))
7574com15 98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → ((𝑣 ∖ {𝑛})𝐺𝐹 → (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
7675com23 83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
7748, 76mpcom 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
7877ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ ℕ0𝑛𝑣) → ((#‘𝑣) = (𝑦 + 1) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
7978com23 83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ ℕ0𝑛𝑣) → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((#‘𝑣) = (𝑦 + 1) → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
8079ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ℕ0 → (𝑛𝑣 → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((#‘𝑣) = (𝑦 + 1) → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
8180com15 98 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑣𝐺𝑒 → (𝑛𝑣 → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
8281imp 443 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣𝐺𝑒𝑛𝑣) → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
8342, 82mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑣𝐺𝑒𝑛𝑣) → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
8483ex 448 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣𝐺𝑒 → (𝑛𝑣 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
8584com4l 89 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝑣 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
8685exlimiv 1844 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑛 𝑛𝑣 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
8741, 86syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
8887ex 448 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ V → (0 < (#‘𝑣) → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
8988com25 96 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ V → (𝑣𝐺𝑒 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
9040, 89ax-mp 5 . . . . . . . . . . . . . . . . . 18 (𝑣𝐺𝑒 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
9190imp 443 . . . . . . . . . . . . . . . . 17 ((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
9291impcom 444 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1))) → (0 < (#‘𝑣) → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))
9339, 92mpd 15 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1))) → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))
9493impancom 454 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ0 ∧ ∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃)) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓))
9594alrimivv 1842 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ0 ∧ ∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃)) → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓))
9695ex 448 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
9728, 96syl5bi 230 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓) → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
986, 10, 14, 18, 20, 97nn0ind 11301 . . . . . . . . . 10 (𝑛 ∈ ℕ0 → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓))
99 brfi1ind.r . . . . . . . . . . . . . 14 Rel 𝐺
10099brrelexi 5069 . . . . . . . . . . . . 13 (𝑉𝐺𝐸𝑉 ∈ V)
10199brrelex2i 5070 . . . . . . . . . . . . 13 (𝑉𝐺𝐸𝐸 ∈ V)
102100, 101jca 552 . . . . . . . . . . . 12 (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
103 breq12 4579 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣𝐺𝑒𝑉𝐺𝐸))
104 fveq2 6085 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
105104eqeq1d 2608 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((#‘𝑣) = 𝑛 ↔ (#‘𝑉) = 𝑛))
106105adantr 479 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑉𝑒 = 𝐸) → ((#‘𝑣) = 𝑛 ↔ (#‘𝑉) = 𝑛))
107103, 106anbi12d 742 . . . . . . . . . . . . . . . 16 ((𝑣 = 𝑉𝑒 = 𝐸) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) ↔ (𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛)))
108 brfi1ind.1 . . . . . . . . . . . . . . . 16 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
109107, 108imbi12d 332 . . . . . . . . . . . . . . 15 ((𝑣 = 𝑉𝑒 = 𝐸) → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) ↔ ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → 𝜑)))
110109spc2gv 3265 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → 𝜑)))
111110com23 83 . . . . . . . . . . . . 13 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑)))
112111expd 450 . . . . . . . . . . . 12 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝐺𝐸 → ((#‘𝑉) = 𝑛 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑))))
113102, 112mpcom 37 . . . . . . . . . . 11 (𝑉𝐺𝐸 → ((#‘𝑉) = 𝑛 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑)))
114113imp 443 . . . . . . . . . 10 ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑))
11598, 114syl5 33 . . . . . . . . 9 ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → (𝑛 ∈ ℕ0𝜑))
116115expcom 449 . . . . . . . 8 ((#‘𝑉) = 𝑛 → (𝑉𝐺𝐸 → (𝑛 ∈ ℕ0𝜑)))
117116com23 83 . . . . . . 7 ((#‘𝑉) = 𝑛 → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸𝜑)))
118117eqcoms 2614 . . . . . 6 (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸𝜑)))
119118imp 443 . . . . 5 ((𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸𝜑))
120119exlimiv 1844 . . . 4 (∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸𝜑))
1212, 120sylbi 205 . . 3 ((#‘𝑉) ∈ ℕ0 → (𝑉𝐺𝐸𝜑))
1221, 121syl 17 . 2 (𝑉 ∈ Fin → (𝑉𝐺𝐸𝜑))
123122impcom 444 1 ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030  wal 1472   = wceq 1474  wex 1694  wcel 1976  Vcvv 3169  cdif 3533  {csn 4121   class class class wbr 4574  Rel wrel 5030  cfv 5787  (class class class)co 6524  Fincfn 7815  cr 9788  0cc0 9789  1c1 9790   + caddc 9792   < clt 9927  0cn0 11136  #chash 12931
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 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
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 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-card 8622  df-cda 8847  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-n0 11137  df-z 11208  df-uz 11517  df-fz 12150  df-hash 12932
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator