Theorem oveq 5894

Description: Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)

Assertion

Ref	Expression
oveq	|- ( F = G -> ( A F B ) = ( A G B ) )

Proof of Theorem oveq

Step	Hyp	Ref	Expression
1		fveq1 5526	. 2 |- ( F = G -> ( F ` <. A , B >. ) = ( G ` <. A , B >. ) )
2		df-ov 5891	. 2 |- ( A F B ) = ( F ` <. A , B >. )
3		df-ov 5891	. 2 |- ( A G B ) = ( G ` <. A , B >. )
4	1, 2, 3	3eqtr4g 2245	1 |- ( F = G -> ( A F B ) = ( A G B ) )

Syntax hints:   -> wi 4   = wceq 1363   <.cop 3607   ` cfv 5228  (class class class)co 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891

This theorem is referenced by:  oveqi  5901  oveqd  5905  ovmpodf  6020  ovmpodv2  6022  mapxpen  6862  ismgm  12795  mgmsscl  12799  issgrp  12828  ismnddef  12841  grpissubg  13094  isrng  13243  islmod  13537  lmodfopne  13572  ispsmet  14176  ismet  14197  isxmet  14198