Theorem tz6.12c 6903

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 6544	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2	1	eqeq1i 2741	. 2 ⊢ ((𝐹‘𝐴) = 𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦)
3		iota1 6513	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
4	2, 3	bitr4id 290	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∃!weu 2568   class class class wbr 5124  ℩cio 6487  ‘cfv 6536

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-ss 3948  df-sn 4607  df-pr 4609  df-uni 4889  df-iota 6489  df-fv 6544

This theorem is referenced by:  tz6.12-1  6904  tz6.12i  6909  fnbrfvb  6934