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Theorem brfi1indALT 13846
Description: Alternate proof of brfi1ind 13845, which does not use brfi1uzind 13844. (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 13705 . . 3 (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
2 dfclel 2891 . . . 4 ((♯‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0))
3 eqeq2 2830 . . . . . . . . . . . . . 14 (𝑥 = 0 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 0))
43anbi2d 628 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0)))
54imbi1d 343 . . . . . . . . . . . 12 (𝑥 = 0 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)))
652albidv 1915 . . . . . . . . . . 11 (𝑥 = 0 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)))
7 eqeq2 2830 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 𝑦))
87anbi2d 628 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦)))
98imbi1d 343 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓)))
1092albidv 1915 . . . . . . . . . . 11 (𝑥 = 𝑦 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓)))
11 eqeq2 2830 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 + 1) → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = (𝑦 + 1)))
1211anbi2d 628 . . . . . . . . . . . . 13 (𝑥 = (𝑦 + 1) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))))
1312imbi1d 343 . . . . . . . . . . . 12 (𝑥 = (𝑦 + 1) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
14132albidv 1915 . . . . . . . . . . 11 (𝑥 = (𝑦 + 1) → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
15 eqeq2 2830 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 𝑛))
1615anbi2d 628 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛)))
1716imbi1d 343 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓)))
18172albidv 1915 . . . . . . . . . . 11 (𝑥 = 𝑛 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓)))
19 brfi1ind.base . . . . . . . . . . . 12 ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)
2019gen2 1788 . . . . . . . . . . 11 𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)
21 breq12 5062 . . . . . . . . . . . . . . 15 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑣𝐺𝑒𝑤𝐺𝑓))
22 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑤 → (♯‘𝑣) = (♯‘𝑤))
2322eqeq1d 2820 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑤 → ((♯‘𝑣) = 𝑦 ↔ (♯‘𝑤) = 𝑦))
2423adantr 481 . . . . . . . . . . . . . . 15 ((𝑣 = 𝑤𝑒 = 𝑓) → ((♯‘𝑣) = 𝑦 ↔ (♯‘𝑤) = 𝑦))
2521, 24anbi12d 630 . . . . . . . . . . . . . 14 ((𝑣 = 𝑤𝑒 = 𝑓) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) ↔ (𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦)))
26 brfi1ind.2 . . . . . . . . . . . . . 14 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
2725, 26imbi12d 346 . . . . . . . . . . . . 13 ((𝑣 = 𝑤𝑒 = 𝑓) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) ↔ ((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)))
2827cbval2vv 2426 . . . . . . . . . . . 12 (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) ↔ ∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃))
29 nn0re 11894 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0𝑦 ∈ ℝ)
30 1re 10629 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℝ
3130a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → 1 ∈ ℝ)
32 nn0ge0 11910 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
33 0lt1 11150 . . . . . . . . . . . . . . . . . . . . 21 0 < 1
3433a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → 0 < 1)
3529, 31, 32, 34addgegt0d 11201 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
3635adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
37 simpr 485 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘𝑣) = (𝑦 + 1))
3836, 37breqtrrd 5085 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (♯‘𝑣))
3938adantrl 712 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))) → 0 < (♯‘𝑣))
40 vex 3495 . . . . . . . . . . . . . . . . . . 19 𝑣 ∈ V
41 hashgt0elex 13750 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣 ∈ V ∧ 0 < (♯‘𝑣)) → ∃𝑛 𝑛𝑣)
42 brfi1ind.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
4340a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑣 ∈ V)
44 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑛𝑣)
45 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑦 ∈ ℕ0)
46 hashdifsnp1 13842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
4743, 44, 45, 46syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑦 ∈ ℕ0𝑛𝑣) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
4847imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦)
49 peano2nn0 11925 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
5049ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → (𝑦 + 1) ∈ ℕ0)
5150ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑦 + 1) ∈ ℕ0)
52 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑣𝐺𝑒)
53 simplrr 774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (♯‘𝑣) = (𝑦 + 1))
54 simprlr 776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) → 𝑛𝑣)
5554adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑛𝑣)
5652, 53, 553jca 1120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
5751, 56jca 512 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
5940, 58ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑣 ∖ {𝑛}) ∈ V
60 brfi1ind.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 𝐹 ∈ V
61 breq12 5062 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑤𝐺𝑓 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
62 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑤 = (𝑣 ∖ {𝑛}) → (♯‘𝑤) = (♯‘(𝑣 ∖ {𝑛})))
6362eqeq1d 2820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑤 = (𝑣 ∖ {𝑛}) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
6463adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
6561, 64anbi12d 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 346 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ↔ (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
6867spc2gv 3598 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ V) → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
6959, 60, 68mp2an 688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
7069expdimp 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦𝜒))
7170ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 680 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦𝜓))
7473exp41 435 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . 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 1922 . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . 17 ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
9291impcom 408 . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ0 ∧ ∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓))
9594alrimivv 1920 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ0 ∧ ∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)) → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓))
9695ex 413 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 → (∀𝑤𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
9728, 96syl5bi 243 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
986, 10, 14, 18, 20, 97nn0ind 12065 . . . . . . . . . 10 (𝑛 ∈ ℕ0 → ∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓))
99 brfi1ind.r . . . . . . . . . . . . . 14 Rel 𝐺
10099brrelex1i 5601 . . . . . . . . . . . . 13 (𝑉𝐺𝐸𝑉 ∈ V)
10199brrelex2i 5602 . . . . . . . . . . . . 13 (𝑉𝐺𝐸𝐸 ∈ V)
102100, 101jca 512 . . . . . . . . . . . 12 (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
103 breq12 5062 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣𝐺𝑒𝑉𝐺𝐸))
104 fveq2 6663 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (♯‘𝑣) = (♯‘𝑉))
105104eqeq1d 2820 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((♯‘𝑣) = 𝑛 ↔ (♯‘𝑉) = 𝑛))
106105adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑉𝑒 = 𝐸) → ((♯‘𝑣) = 𝑛 ↔ (♯‘𝑉) = 𝑛))
107103, 106anbi12d 630 . . . . . . . . . . . . . . . 16 ((𝑣 = 𝑉𝑒 = 𝐸) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) ↔ (𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛)))
108 brfi1ind.1 . . . . . . . . . . . . . . . 16 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
109107, 108imbi12d 346 . . . . . . . . . . . . . . 15 ((𝑣 = 𝑉𝑒 = 𝐸) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) ↔ ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → 𝜑)))
110109spc2gv 3598 . . . . . . . . . . . . . 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 416 . . . . . . . . . . . 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 407 . . . . . . . . . 10 ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → (∀𝑣𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) → 𝜑))
11598, 114syl5 34 . . . . . . . . 9 ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → (𝑛 ∈ ℕ0𝜑))
116115expcom 414 . . . . . . . 8 ((♯‘𝑉) = 𝑛 → (𝑉𝐺𝐸 → (𝑛 ∈ ℕ0𝜑)))
117116com23 86 . . . . . . 7 ((♯‘𝑉) = 𝑛 → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸𝜑)))
118117eqcoms 2826 . . . . . 6 (𝑛 = (♯‘𝑉) → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸𝜑)))
119118imp 407 . . . . 5 ((𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸𝜑))
120119exlimiv 1922 . . . 4 (∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸𝜑))
1212, 120sylbi 218 . . 3 ((♯‘𝑉) ∈ ℕ0 → (𝑉𝐺𝐸𝜑))
1221, 121syl 17 . 2 (𝑉 ∈ Fin → (𝑉𝐺𝐸𝜑))
123122impcom 408 1 ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079  wal 1526   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  cdif 3930  {csn 4557   class class class wbr 5057  Rel wrel 5553  cfv 6348  (class class class)co 7145  Fincfn 8497  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  0cn0 11885  chash 13678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12881  df-hash 13679
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator