Theorem oveq 5848

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 5485	. 2 |- ( F = G -> ( F ` <. A , B >. ) = ( G ` <. A , B >. ) )
2		df-ov 5845	. 2 |- ( A F B ) = ( F ` <. A , B >. )
3		df-ov 5845	. 2 |- ( A G B ) = ( G ` <. A , B >. )
4	1, 2, 3	3eqtr4g 2224	1 |- ( F = G -> ( A F B ) = ( A G B ) )

Syntax hints:   -> wi 4   = wceq 1343   <.cop 3579   ` cfv 5188  (class class class)co 5842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  oveqi  5855  oveqd  5859  ovmpodf  5973  ovmpodv2  5975  mapxpen  6814  ismgm  12588  mgmsscl  12592  ispsmet  12963  ismet  12984  isxmet  12985