Theorem tz6.12c-afv2 47254

Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6928. (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 2588	. . . 4 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦
2		nfv 1914	. . . 4 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹''''𝐴)
3		euex 2577	. . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
4		tz6.12-1-afv2 47253	. . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)
5	4	expcom 413	. . . . 5 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹''''𝐴) = 𝑦))
6		breq2 5147	. . . . . 6 ⊢ ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
7	6	biimprd 248	. . . . 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 2219	. . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹''''𝐴))
10	9, 6	syl5ibcom 245	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 → 𝐴𝐹𝑦))
11	10, 5	impbid 212	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∃!weu 2568   class class class wbr 5143  ''''cafv2 47220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-dfat 47131  df-afv2 47221

This theorem is referenced by:  tz6.12i-afv2  47255  dfatbrafv2b  47257  fnbrafv2b  47260  dfatcolem  47267