Theorem oveq 5973

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 5598	. 2 |- ( F = G -> ( F ` <. A , B >. ) = ( G ` <. A , B >. ) )
2		df-ov 5970	. 2 |- ( A F B ) = ( F ` <. A , B >. )
3		df-ov 5970	. 2 |- ( A G B ) = ( G ` <. A , B >. )
4	1, 2, 3	3eqtr4g 2265	1 |- ( F = G -> ( A F B ) = ( A G B ) )

Syntax hints:   -> wi 4   = wceq 1373   <.cop 3646   ` cfv 5290  (class class class)co 5967

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  oveqi  5980  oveqd  5984  ovmpodf  6100  ovmpodv2  6102  mapxpen  6970  ismgm  13304  mgmsscl  13308  issgrp  13350  ismnddef  13365  grpissubg  13645  isrng  13811  islmod  14168  lmodfopne  14203  ispsmet  14910  ismet  14931  isxmet  14932