Theorem tz6.12c-afv2 48585

Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6938. (Contributed by AV, 5-Sep-2022.)

Assertion

Ref	Expression
tz6.12c-afv2	⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))

Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12c-afv2

Step	Hyp	Ref	Expression
1		nfeu1 2617	. . . 4 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦
2		nfv 1949	. . . 4 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹''''𝐴)
3		euex 2605	. . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
4		tz6.12-1-afv2 48584	. . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
5	4	expcom 419	. . . . 5 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹''''𝐴) = 𝑦))
6		breq2 5112	. . . . . 6 ⊢ ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
7	6	biimprd 251	. . . . 5 ⊢ ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹''''𝐴)))
8	5, 7	syli 40	. . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹''''𝐴)))
9	1, 2, 3, 8	exlimimdd 2259	. . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹''''𝐴))
10	9, 6	syl5ibcom 248	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 → 𝐴𝐹𝑦))
11	10, 5	impbid 215	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃!weu 2596   class class class wbr 5108  ''''cafv2 48551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-res 5670  df-iota 6520  df-fun 6568  df-fn 6569  df-dfat 48463  df-afv2 48552

This theorem is used by:  tz6.12i-afv2  48586  dfatbrafv2b  48588  fnbrafv2b  48591  dfatcolem  48598