Theorem fveq1 5197

Description: Equality theorem for function value. (Contributed by NM, 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 3787	. . 3 |- ( F = G -> ( A F x <-> A G x ) )
2	1	iotabidv 4908	. 2 |- ( F = G -> ( iota x A F x ) = ( iota x A G x ) )
3		df-fv 4930	. 2 |- ( F ` A ) = ( iota x A F x )
4		df-fv 4930	. 2 |- ( G ` A ) = ( iota x A G x )
5	2, 3, 4	3eqtr4g 2138	1 |- ( F = G -> ( F ` A ) = ( G ` A ) )

Syntax hints:   -> wi 4   = wceq 1284   class class class wbr 3785   iotacio 4885   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930

This theorem is referenced by:  fveq1i  5199  fveq1d  5200  fvmptdf  5279  fvmptdv2  5281  isoeq1  5461  oveq  5538  offval  5739  ofrfval  5740  offval3  5781  smoeq  5928  recseq  5944  tfr0  5960  tfrlemiex  5968  rdgeq1  5981  rdgivallem  5991  rdg0  5997  frec0g  6006  frecsuclem3  6013  frecsuc  6014  ac6sfi  6379  1fv  9149  iseqeq3  9436  shftvalg  9724  shftval4g  9725  clim  10120