Theorem fveq1 5328

Description: Equality theorem for function value. (Contributed by set.mm contributors, 29-Dec-1996.)

Assertion

Ref	Expression
fveq1	⊢ (F = G → (F ‘A) = (G ‘A))

Proof of Theorem fveq1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 4642	. . 3 ⊢ (F = G → (AFx ↔ AGx))
2	1	iotabidv 4361	. 2 ⊢ (F = G → (℩xAFx) = (℩xAGx))
3		df-fv 4796	. 2 ⊢ (F ‘A) = (℩xAFx)
4		df-fv 4796	. 2 ⊢ (G ‘A) = (℩xAGx)
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (F = G → (F ‘A) = (G ‘A))

Syntax hints:   → wi 4   = wceq 1642  ℩cio 4338   class class class wbr 4640   ‘cfv 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-uni 3893  df-iota 4340  df-br 4641  df-fv 4796

This theorem is referenced by:  fveq1i  5330  fveq1d  5331  eqfnfv  5393  isoeq1  5483  oveq  5530