Theorem fveq1 5327

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 4641	. . 3 |- (F = G -> (AFx <-> AGx))
2	1	iotabidv 4360	. 2 |- (F = G -> (iotaxAFx) = (iotaxAGx))
3		df-fv 4795	. 2 |- (F` A) = (iotaxAFx)
4		df-fv 4795	. 2 |- (G` A) = (iotaxAGx)
5	2, 3, 4	3eqtr4g 2410	1 |- (F = G -> (F` A) = (G` A))

Syntax hints:   -> wi 4   = wceq 1642  iotacio 4337   class class class wbr 4639  ` cfv 4781

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 2478  df-rex 2620  df-uni 3892  df-iota 4339  df-br 4640  df-fv 4795

This theorem is referenced by:  fveq1i  5329  fveq1d  5330  eqfnfv  5392  isoeq1  5482  oveq  5529