Theorem tz6.12c 6907

Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by SN, 23-Dec-2024.)

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		df-fv 6548	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2	1	eqeq1i 2770	. 2 ⊢ ((𝐹‘𝐴) = 𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦)
3		iota1 6519	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
4	2, 3	bitr4id 293	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃!weu 2598   class class class wbr 5111  ℩cio 6494  ‘cfv 6540

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-ss 3923  df-sn 4592  df-pr 4594  df-uni 4875  df-iota 6496  df-fv 6548

This theorem is used by:  tz6.12-1  6908  tz6.12i  6911  fnbrfvb  6935