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Theorem fveq1 5204
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.
StepHypRef Expression
1 breq 3793 . . 3  |-  ( F  =  G  ->  ( A F x  <->  A G x ) )
21iotabidv 4915 . 2  |-  ( F  =  G  ->  ( iota x A F x )  =  ( iota
x A G x ) )
3 df-fv 4937 . 2  |-  ( F `
 A )  =  ( iota x A F x )
4 df-fv 4937 . 2  |-  ( G `
 A )  =  ( iota x A G x )
52, 3, 43eqtr4g 2113 1  |-  ( F  =  G  ->  ( F `  A )  =  ( G `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259   class class class wbr 3791   iotacio 4892   ` cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-uni 3608  df-br 3792  df-iota 4894  df-fv 4937
This theorem is referenced by:  fveq1i  5206  fveq1d  5207  fvmptdf  5285  fvmptdv2  5287  isoeq1  5468  oveq  5545  offval  5746  ofrfval  5747  offval3  5788  smoeq  5935  recseq  5951  tfr0  5967  tfrlemiex  5975  rdgeq1  5988  rdgivallem  5998  rdg0  6004  frec0g  6013  frecsuclem3  6020  frecsuc  6021  ac6sfi  6382  1fv  9097  iseqeq3  9379  shftvalg  9664  shftval4g  9665  clim  10032
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