Theorem tz6.12c 6901

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 6541	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2	1	eqeq1i 2774	. 2 ⊢ ((𝐹‘𝐴) = 𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦)
3		iota1 6512	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
4	2, 3	bitr4id 293	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567  ∃!weu 2602   class class class wbr 5110  ℩cio 6487  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-sn 4592  df-pr 4594  df-uni 4874  df-iota 6489  df-fv 6541

This theorem is referenced by:  tz6.12-1  6902  tz6.12i  6905  fnbrfvb  6929