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Theorem brfi1indALT 13320
Description: Alternate proof of brfi1ind 13319, which does not use brfi1uzind 13318. (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 13185 . . 3 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
2 df-clel 2647 . . . 4 ((#‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0))
3 eqeq2 2662 . . . . . . . . . . . . . 14 (𝑥 = 0 → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = 0))
43anbi2d 740 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = 0)))
54imbi1d 330 . . . . . . . . . . . 12 (𝑥 = 0 → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)))
652albidv 1891 . . . . . . . . . . 11 (𝑥 = 0 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)))
7 eqeq2 2662 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = 𝑦))
87anbi2d 740 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦)))
98imbi1d 330 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓)))
1092albidv 1891 . . . . . . . . . . 11 (𝑥 = 𝑦 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓)))
11 eqeq2 2662 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 + 1) → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = (𝑦 + 1)))
1211anbi2d 740 . . . . . . . . . . . . 13 (𝑥 = (𝑦 + 1) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1))))
1312imbi1d 330 . . . . . . . . . . . 12 (𝑥 = (𝑦 + 1) → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
14132albidv 1891 . . . . . . . . . . 11 (𝑥 = (𝑦 + 1) → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
15 eqeq2 2662 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = 𝑛))
1615anbi2d 740 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛)))
1716imbi1d 330 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓)))
18172albidv 1891 . . . . . . . . . . 11 (𝑥 = 𝑛 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓)))
19 brfi1ind.base . . . . . . . . . . . 12 ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)
2019gen2 1763 . . . . . . . . . . 11 𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)
21 breq12 4690 . . . . . . . . . . . . . . 15 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑣𝐺𝑒𝑤𝐺𝑓))
22 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
2322eqeq1d 2653 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑤 → ((#‘𝑣) = 𝑦 ↔ (#‘𝑤) = 𝑦))
2423adantr 480 . . . . . . . . . . . . . . 15 ((𝑣 = 𝑤𝑒 = 𝑓) → ((#‘𝑣) = 𝑦 ↔ (#‘𝑤) = 𝑦))
2521, 24anbi12d 747 . . . . . . . . . . . . . 14 ((𝑣 = 𝑤𝑒 = 𝑓) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) ↔ (𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦)))
26 brfi1ind.2 . . . . . . . . . . . . . 14 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
2725, 26imbi12d 333 . . . . . . . . . . . . 13 ((𝑣 = 𝑤𝑒 = 𝑓) → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓) ↔ ((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃)))
2827cbval2v 2321 . . . . . . . . . . . 12 (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓) ↔ ∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃))
29 nn0re 11339 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0𝑦 ∈ ℝ)
30 1re 10077 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℝ
3130a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → 1 ∈ ℝ)
32 nn0ge0 11356 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
33 0lt1 10588 . . . . . . . . . . . . . . . . . . . . 21 0 < 1
3433a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → 0 < 1)
3529, 31, 32, 34addgegt0d 10639 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
3635adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
37 simpr 476 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → (#‘𝑣) = (𝑦 + 1))
3836, 37breqtrrd 4713 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (#‘𝑣))
3938adantrl 752 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1))) → 0 < (#‘𝑣))
40 vex 3234 . . . . . . . . . . . . . . . . . . 19 𝑣 ∈ V
41 hashgt0elex 13227 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ∃𝑛 𝑛𝑣)
42 brfi1ind.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
4340a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑣 ∈ V)
44 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑛𝑣)
45 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑦 ∈ ℕ0)
46 brfi1indlem 13316 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0) → ((#‘𝑣) = (𝑦 + 1) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
4743, 44, 45, 46syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑦 ∈ ℕ0𝑛𝑣) → ((#‘𝑣) = (𝑦 + 1) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
4847imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (#‘(𝑣 ∖ {𝑛})) = 𝑦)
49 peano2nn0 11371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
5049ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑦 + 1) ∈ ℕ0)
5150ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑦 + 1) ∈ ℕ0)
52 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑣𝐺𝑒)
53 simplrr 818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (#‘𝑣) = (𝑦 + 1))
54 simprlr 820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) → 𝑛𝑣)
5554adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑛𝑣)
5652, 53, 553jca 1261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
5751, 56jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
5940, 58ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑣 ∖ {𝑛}) ∈ V
60 brfi1ind.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 𝐹 ∈ V
61 breq12 4690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑤𝐺𝑓 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
62 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
6362eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑤 = (𝑣 ∖ {𝑛}) → ((#‘𝑤) = 𝑦 ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
6463adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((#‘𝑤) = 𝑦 ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
6561, 64anbi12d 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ↔ (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
6867spc2gv 3327 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ V) → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
6959, 60, 68mp2an 708 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
7069expdimp 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜒))
7170ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜓))
7473exp41 637 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑣𝐺𝑒 → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜓)))))
7574com15 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → ((𝑣 ∖ {𝑛})𝐺𝐹 → (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
7675com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
7748, 76mpcom 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
7877ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ ℕ0𝑛𝑣) → ((#‘𝑣) = (𝑦 + 1) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
7978com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ ℕ0𝑛𝑣) → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((#‘𝑣) = (𝑦 + 1) → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
8079ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ℕ0 → (𝑛𝑣 → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((#‘𝑣) = (𝑦 + 1) → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
8180com15 101 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑣𝐺𝑒 → (𝑛𝑣 → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
8281imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣𝐺𝑒 → (𝑛𝑣 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
8584com4l 92 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝑣 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
8685exlimiv 1898 . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ V → (0 < (#‘𝑣) → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
8988com25 99 . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . 17 ((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
9291impcom 445 . . . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ0 ∧ ∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃)) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓))
9594alrimivv 1896 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ0 ∧ ∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃)) → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓))
9695ex 449 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
9728, 96syl5bi 232 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓) → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
986, 10, 14, 18, 20, 97nn0ind 11510 . . . . . . . . . 10 (𝑛 ∈ ℕ0 → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓))
99 brfi1ind.r . . . . . . . . . . . . . 14 Rel 𝐺
10099brrelexi 5192 . . . . . . . . . . . . 13 (𝑉𝐺𝐸𝑉 ∈ V)
10199brrelex2i 5193 . . . . . . . . . . . . 13 (𝑉𝐺𝐸𝐸 ∈ V)
102100, 101jca 553 . . . . . . . . . . . 12 (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
103 breq12 4690 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣𝐺𝑒𝑉𝐺𝐸))
104 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
105104eqeq1d 2653 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((#‘𝑣) = 𝑛 ↔ (#‘𝑉) = 𝑛))
106105adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑉𝑒 = 𝐸) → ((#‘𝑣) = 𝑛 ↔ (#‘𝑉) = 𝑛))
107103, 106anbi12d 747 . . . . . . . . . . . . . . . 16 ((𝑣 = 𝑉𝑒 = 𝐸) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) ↔ (𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛)))
108 brfi1ind.1 . . . . . . . . . . . . . . . 16 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
109107, 108imbi12d 333 . . . . . . . . . . . . . . 15 ((𝑣 = 𝑉𝑒 = 𝐸) → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) ↔ ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → 𝜑)))
110109spc2gv 3327 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → 𝜑)))
111110com23 86 . . . . . . . . . . . . 13 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑)))
112111expd 451 . . . . . . . . . . . 12 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝐺𝐸 → ((#‘𝑉) = 𝑛 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑))))
113102, 112mpcom 38 . . . . . . . . . . 11 (𝑉𝐺𝐸 → ((#‘𝑉) = 𝑛 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑)))
114113imp 444 . . . . . . . . . 10 ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑))
11598, 114syl5 34 . . . . . . . . 9 ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → (𝑛 ∈ ℕ0𝜑))
116115expcom 450 . . . . . . . 8 ((#‘𝑉) = 𝑛 → (𝑉𝐺𝐸 → (𝑛 ∈ ℕ0𝜑)))
117116com23 86 . . . . . . 7 ((#‘𝑉) = 𝑛 → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸𝜑)))
118117eqcoms 2659 . . . . . 6 (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸𝜑)))
119118imp 444 . . . . 5 ((𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸𝜑))
120119exlimiv 1898 . . . 4 (∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸𝜑))
1212, 120sylbi 207 . . 3 ((#‘𝑉) ∈ ℕ0 → (𝑉𝐺𝐸𝜑))
1221, 121syl 17 . 2 (𝑉 ∈ Fin → (𝑉𝐺𝐸𝜑))
123122impcom 445 1 ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cdif 3604  {csn 4210   class class class wbr 4685  Rel wrel 5148  cfv 5926  (class class class)co 6690  Fincfn 7997  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  0cn0 11330  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator