Theorem tz6.12c 6781

Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Assertion

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

Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12c

Step	Hyp	Ref	Expression
1		nfeu1 2588	. . . 4 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦
2		nfv 1918	. . . 4 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹‘𝐴)
3		euex 2577	. . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
4		tz6.12-1 6778	. . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
5	4	expcom 413	. . . . 5 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹‘𝐴) = 𝑦))
6		breq2 5074	. . . . . 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 2215	. . 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 1539  ∃!weu 2568   class class class wbr 5070  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426

This theorem is referenced by:  tz6.12i  6782  fnbrfvb  6804