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Theorem fi1uzindOLD 13140
 Description: Obsolete version of fi1uzind 13134 as of 28-Mar-2021. (Contributed by AV, 22-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
fi1uzindOLD.f 𝐹𝑈
fi1uzindOLD.l 𝐿 ∈ ℕ0
fi1uzindOLD.1 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
fi1uzindOLD.2 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
fi1uzindOLD.3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)
fi1uzindOLD.4 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
fi1uzindOLD.base (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)
fi1uzindOLD.step ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
Assertion
Ref Expression
fi1uzindOLD (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
Distinct variable groups:   𝑎,𝑏,𝑒,𝑛,𝑣,𝑦,𝑓,𝑤   𝐸,𝑎,𝑒,𝑛,𝑣   𝐹,𝑎,𝑓,𝑤   𝑒,𝑓,𝑤,𝑛,𝑣,𝑦   𝑒,𝐿,𝑛,𝑣,𝑦   𝑉,𝑎,𝑏,𝑒,𝑛,𝑣   𝜓,𝑓,𝑛,𝑤,𝑦   𝜃,𝑒,𝑛,𝑣   𝜒,𝑓,𝑤   𝜑,𝑒,𝑛,𝑣   𝜌,𝑒,𝑓,𝑛,𝑣,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑦,𝑤,𝑓,𝑎,𝑏)   𝜓(𝑣,𝑒,𝑎,𝑏)   𝜒(𝑦,𝑣,𝑒,𝑛,𝑎,𝑏)   𝜃(𝑦,𝑤,𝑓,𝑎,𝑏)   𝜌(𝑎,𝑏)   𝑈(𝑦,𝑤,𝑣,𝑒,𝑓,𝑛,𝑎,𝑏)   𝐸(𝑦,𝑤,𝑓,𝑏)   𝐹(𝑦,𝑣,𝑒,𝑛,𝑏)   𝐿(𝑤,𝑓,𝑎,𝑏)   𝑉(𝑦,𝑤,𝑓)

Proof of Theorem fi1uzindOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hashcl 13009 . . . 4 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
2 df-clel 2606 . . . . 5 ((#‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0))
3 fi1uzindOLD.l . . . . . . . . . . . . . . 15 𝐿 ∈ ℕ0
4 nn0z 11277 . . . . . . . . . . . . . . 15 (𝐿 ∈ ℕ0𝐿 ∈ ℤ)
53, 4mp1i 13 . . . . . . . . . . . . . 14 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝐿 ∈ ℤ)
6 nn0z 11277 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0𝑛 ∈ ℤ)
76ad2antlr 759 . . . . . . . . . . . . . 14 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝑛 ∈ ℤ)
8 breq2 4587 . . . . . . . . . . . . . . . . . 18 ((#‘𝑉) = 𝑛 → (𝐿 ≤ (#‘𝑉) ↔ 𝐿𝑛))
98eqcoms 2618 . . . . . . . . . . . . . . . . 17 (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) ↔ 𝐿𝑛))
109biimpcd 238 . . . . . . . . . . . . . . . 16 (𝐿 ≤ (#‘𝑉) → (𝑛 = (#‘𝑉) → 𝐿𝑛))
1110adantr 480 . . . . . . . . . . . . . . 15 ((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑛 = (#‘𝑉) → 𝐿𝑛))
1211imp 444 . . . . . . . . . . . . . 14 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝐿𝑛)
13 eqeq1 2614 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐿 → (𝑥 = (#‘𝑣) ↔ 𝐿 = (#‘𝑣)))
1413anbi2d 736 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐿 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣))))
1514imbi1d 330 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐿 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)))
16152albidv 1838 . . . . . . . . . . . . . . 15 (𝑥 = 𝐿 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)))
17 eqeq1 2614 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (𝑥 = (#‘𝑣) ↔ 𝑦 = (#‘𝑣)))
1817anbi2d 736 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣))))
1918imbi1d 330 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓)))
20192albidv 1838 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓)))
21 eqeq1 2614 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑦 + 1) → (𝑥 = (#‘𝑣) ↔ (𝑦 + 1) = (#‘𝑣)))
2221anbi2d 736 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 + 1) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))))
2322imbi1d 330 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 + 1) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
24232albidv 1838 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 + 1) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
25 eqeq1 2614 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑛 → (𝑥 = (#‘𝑣) ↔ 𝑛 = (#‘𝑣)))
2625anbi2d 736 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑛 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣))))
2726imbi1d 330 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑛 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓)))
28272albidv 1838 . . . . . . . . . . . . . . 15 (𝑥 = 𝑛 → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓)))
29 eqcom 2617 . . . . . . . . . . . . . . . . . 18 (𝐿 = (#‘𝑣) ↔ (#‘𝑣) = 𝐿)
30 fi1uzindOLD.base . . . . . . . . . . . . . . . . . 18 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)
3129, 30sylan2b 491 . . . . . . . . . . . . . . . . 17 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)
3231gen2 1714 . . . . . . . . . . . . . . . 16 𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓)
3332a1i 11 . . . . . . . . . . . . . . 15 (𝐿 ∈ ℤ → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝐿 = (#‘𝑣)) → 𝜓))
34 simpl 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑤𝑒 = 𝑓) → 𝑣 = 𝑤)
35 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 = 𝑤𝑒 = 𝑓) → 𝑒 = 𝑓)
3635sbceq1d 3407 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑤𝑒 = 𝑓) → ([𝑒 / 𝑏]𝜌[𝑓 / 𝑏]𝜌))
3734, 36sbceqbid 3409 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑤𝑒 = 𝑓) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌[𝑤 / 𝑎][𝑓 / 𝑏]𝜌))
38 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
3938eqeq2d 2620 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑤 → (𝑦 = (#‘𝑣) ↔ 𝑦 = (#‘𝑤)))
4039adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑦 = (#‘𝑣) ↔ 𝑦 = (#‘𝑤)))
4137, 40anbi12d 743 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑤𝑒 = 𝑓) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) ↔ ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤))))
42 fi1uzindOLD.2 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
4341, 42imbi12d 333 . . . . . . . . . . . . . . . . 17 ((𝑣 = 𝑤𝑒 = 𝑓) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓) ↔ (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃)))
4443cbval2v 2273 . . . . . . . . . . . . . . . 16 (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓) ↔ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃))
45 nn0ge0 11195 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0 → 0 ≤ 𝐿)
46 0red 9920 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℤ → 0 ∈ ℝ)
47 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
483, 47mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℤ → 𝐿 ∈ ℝ)
49 zre 11258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
50 letr 10010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 ≤ 𝐿𝐿𝑦) → 0 ≤ 𝑦))
5146, 48, 49, 50syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ℤ → ((0 ≤ 𝐿𝐿𝑦) → 0 ≤ 𝑦))
52 0nn0 11184 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 ∈ ℕ0
53 pm3.22 464 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((0 ≤ 𝑦𝑦 ∈ ℤ) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
54 0z 11265 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 0 ∈ ℤ
55 eluz1 11567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0 ∈ ℤ → (𝑦 ∈ (ℤ‘0) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)))
5654, 55mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((0 ≤ 𝑦𝑦 ∈ ℤ) → (𝑦 ∈ (ℤ‘0) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)))
5753, 56mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((0 ≤ 𝑦𝑦 ∈ ℤ) → 𝑦 ∈ (ℤ‘0))
58 eluznn0 11633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((0 ∈ ℕ0𝑦 ∈ (ℤ‘0)) → 𝑦 ∈ ℕ0)
5952, 57, 58sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0 ≤ 𝑦𝑦 ∈ ℤ) → 𝑦 ∈ ℕ0)
6059ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 ≤ 𝑦 → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0))
6151, 60syl6com 36 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ≤ 𝐿𝐿𝑦) → (𝑦 ∈ ℤ → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0)))
6261ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 ≤ 𝐿 → (𝐿𝑦 → (𝑦 ∈ ℤ → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0))))
6362com14 94 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℤ → (𝐿𝑦 → (𝑦 ∈ ℤ → (0 ≤ 𝐿𝑦 ∈ ℕ0))))
6463pm2.43a 52 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℤ → (𝐿𝑦 → (0 ≤ 𝐿𝑦 ∈ ℕ0)))
6564imp 444 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℤ ∧ 𝐿𝑦) → (0 ≤ 𝐿𝑦 ∈ ℕ0))
6665com12 32 . . . . . . . . . . . . . . . . . . . . . 22 (0 ≤ 𝐿 → ((𝑦 ∈ ℤ ∧ 𝐿𝑦) → 𝑦 ∈ ℕ0))
673, 45, 66mp2b 10 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℤ ∧ 𝐿𝑦) → 𝑦 ∈ ℕ0)
68673adant1 1072 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → 𝑦 ∈ ℕ0)
69 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 + 1) = (#‘𝑣) ↔ (#‘𝑣) = (𝑦 + 1))
70 nn0p1gt0 11199 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
7170adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
72 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → (#‘𝑣) = (𝑦 + 1))
7371, 72breqtrrd 4611 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (#‘𝑣))
7469, 73sylan2b 491 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℕ0 ∧ (𝑦 + 1) = (#‘𝑣)) → 0 < (#‘𝑣))
7574adantrl 748 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → 0 < (#‘𝑣))
76 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑣 ∈ V
77 hashgt0elex 13050 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ∃𝑛 𝑛𝑣)
78 fi1uzindOLD.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)
7976a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑣 ∈ V)
80 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑛𝑣)
81 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑦 ∈ ℕ0𝑛𝑣) → 𝑦 ∈ ℕ0)
82 brfi1indlem 13133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0) → ((#‘𝑣) = (𝑦 + 1) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
8369, 82syl5bi 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑣 ∈ V ∧ 𝑛𝑣𝑦 ∈ ℕ0) → ((𝑦 + 1) = (#‘𝑣) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
8479, 80, 81, 83syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑦 ∈ ℕ0𝑛𝑣) → ((𝑦 + 1) = (#‘𝑣) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
8584imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → (#‘(𝑣 ∖ {𝑛})) = 𝑦)
86 peano2nn0 11210 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
8786ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → (𝑦 + 1) ∈ ℕ0)
8887ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → (𝑦 + 1) ∈ ℕ0)
89 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → [𝑣 / 𝑎][𝑒 / 𝑏]𝜌)
90 simplrr 797 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → (𝑦 + 1) = (#‘𝑣))
91 simprlr 799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) → 𝑛𝑣)
9291adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → 𝑛𝑣)
9389, 90, 923jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣))
9488, 93jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → ((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣)))
95 difexg 4735 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
9676, 95ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑣 ∖ {𝑛}) ∈ V
97 fi1uzindOLD.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 𝐹𝑈
98 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑤 = (𝑣 ∖ {𝑛}))
99 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
10099sbceq1d 3407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑓 / 𝑏]𝜌[𝐹 / 𝑏]𝜌))
10198, 100sbceqbid 3409 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌[(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌))
102 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 (𝑦 = (#‘𝑤) ↔ (#‘𝑤) = 𝑦)
103 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
104103eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 (𝑤 = (𝑣 ∖ {𝑛}) → ((#‘𝑤) = 𝑦 ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
105102, 104syl5bb 271 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (𝑤 = (𝑣 ∖ {𝑛}) → (𝑦 = (#‘𝑤) ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
106105adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑦 = (#‘𝑤) ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
107101, 106anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) ↔ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦)))
108 fi1uzindOLD.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
109107, 108imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ↔ (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
110109spc2gv 3269 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹𝑈) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
11196, 97, 110mp2an 704 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
112111expdimp 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜒))
113112ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜒))
114693anbi2i 1247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
115114anbi2i 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣)) ↔ ((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)))
116 fi1uzindOLD.step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
117115, 116sylanb 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
11894, 113, 117syl6an 566 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜓))
119118exp41 636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((#‘(𝑣 ∖ {𝑛})) = 𝑦𝜓)))))
120119com15 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
121120com23 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
12285, 121mpcom 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑦 ∈ ℕ0𝑛𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))
123122ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑦 ∈ ℕ0𝑛𝑣) → ((𝑦 + 1) = (#‘𝑣) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
124123com23 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑦 ∈ ℕ0𝑛𝑣) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
125124ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 ∈ ℕ0 → (𝑛𝑣 → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
126125com15 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (𝑛𝑣 → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
127126imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
12878, 127mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))
129128ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (𝑛𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
130129com4l 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
131130exlimiv 1845 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃𝑛 𝑛𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
13277, 131syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
133132ex 449 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 ∈ V → (0 < (#‘𝑣) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
134133com25 97 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 ∈ V → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
13576, 134ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
136135imp 444 . . . . . . . . . . . . . . . . . . . . . 22 (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))
137136impcom 445 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (0 < (#‘𝑣) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))
13875, 137mpd 15 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))
13968, 138sylan 487 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))
140139impancom 455 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃)) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓))
141140alrimivv 1843 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) ∧ ∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃)) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓))
142141ex 449 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → (∀𝑤𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌𝑦 = (#‘𝑤)) → 𝜃) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
14344, 142syl5bi 231 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿𝑦) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑦 = (#‘𝑣)) → 𝜓) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
14416, 20, 24, 28, 33, 143uzind 11345 . . . . . . . . . . . . . 14 ((𝐿 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐿𝑛) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓))
1455, 7, 12, 144syl3anc 1318 . . . . . . . . . . . . 13 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → ∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓))
146 sbcex 3412 . . . . . . . . . . . . . . . 16 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ V)
147 sbccom 3476 . . . . . . . . . . . . . . . . 17 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌[𝐸 / 𝑏][𝑉 / 𝑎]𝜌)
148 sbcex 3412 . . . . . . . . . . . . . . . . 17 ([𝐸 / 𝑏][𝑉 / 𝑎]𝜌𝐸 ∈ V)
149147, 148sylbi 206 . . . . . . . . . . . . . . . 16 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝐸 ∈ V)
150146, 149jca 553 . . . . . . . . . . . . . . 15 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
151 simpl 472 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
152 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑒 = 𝐸)
153152sbceq1d 3407 . . . . . . . . . . . . . . . . . . . . 21 ((𝑣 = 𝑉𝑒 = 𝐸) → ([𝑒 / 𝑏]𝜌[𝐸 / 𝑏]𝜌))
154151, 153sbceqbid 3409 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑉𝑒 = 𝐸) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌[𝑉 / 𝑎][𝐸 / 𝑏]𝜌))
155 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
156155eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑉 → (𝑛 = (#‘𝑣) ↔ 𝑛 = (#‘𝑉)))
157156adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑛 = (#‘𝑣) ↔ 𝑛 = (#‘𝑉)))
158154, 157anbi12d 743 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑉𝑒 = 𝐸) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) ↔ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉))))
159 fi1uzindOLD.1 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
160158, 159imbi12d 333 . . . . . . . . . . . . . . . . . 18 ((𝑣 = 𝑉𝑒 = 𝐸) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) ↔ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑)))
161160spc2gv 3269 . . . . . . . . . . . . . . . . 17 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑)))
162161com23 84 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑)))
163162expd 451 . . . . . . . . . . . . . . 15 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (#‘𝑉) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑))))
164150, 163mpcom 37 . . . . . . . . . . . . . 14 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (#‘𝑉) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑)))
165164imp 444 . . . . . . . . . . . . 13 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → (∀𝑣𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛 = (#‘𝑣)) → 𝜓) → 𝜑))
166145, 165syl5com 31 . . . . . . . . . . . 12 (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑))
167166exp31 628 . . . . . . . . . . 11 (𝐿 ≤ (#‘𝑉) → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → 𝜑))))
168167com14 94 . . . . . . . . . 10 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑛 = (#‘𝑉)) → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) → 𝜑))))
169168expcom 450 . . . . . . . . 9 (𝑛 = (#‘𝑉) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) → 𝜑)))))
170169com24 93 . . . . . . . 8 (𝑛 = (#‘𝑉) → (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))))
171170pm2.43i 50 . . . . . . 7 (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑))))
172171imp 444 . . . . . 6 ((𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
173172exlimiv 1845 . . . . 5 (∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
1742, 173sylbi 206 . . . 4 ((#‘𝑉) ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
1751, 174syl 17 . . 3 (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
176175com12 32 . 2 ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ Fin → (𝐿 ≤ (#‘𝑉) → 𝜑)))
1771763imp 1249 1 (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  [wsbc 3402   ∖ cdif 3537  {csn 4125   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  #chash 12979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980 This theorem is referenced by:  brfi1uzindOLD  13141  opfi1uzindOLD  13144
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