Theorem tz6.12c-afv2 46684

Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6913. (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 2576	. . . 4 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦
2		nfv 1909	. . . 4 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹''''𝐴)
3		euex 2565	. . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
4		tz6.12-1-afv2 46683	. . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
5	4	expcom 412	. . . . 5 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹''''𝐴) = 𝑦))
6		breq2 5147	. . . . . 6 ⊢ ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
7	6	biimprd 247	. . . . 5 ⊢ ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹''''𝐴)))
8	5, 7	syli 39	. . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹''''𝐴)))
9	1, 2, 3, 8	exlimimdd 2207	. . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹''''𝐴))
10	9, 6	syl5ibcom 244	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 → 𝐴𝐹𝑦))
11	10, 5	impbid 211	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533  ∃!weu 2556   class class class wbr 5143  ''''cafv2 46650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6494  df-fun 6544  df-fn 6545  df-dfat 46561  df-afv2 46651

This theorem is referenced by:  tz6.12i-afv2  46685  dfatbrafv2b  46687  fnbrafv2b  46690  dfatcolem  46697