Theorem oprssov 6231

Description: The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.)

Assertion

Ref	Expression
oprssov	|- ( ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) /\ ( A e. C /\ B e. D ) ) -> ( A F B ) = ( A G B ) )

Proof of Theorem oprssov

Step	Hyp	Ref	Expression
1		ovres 6229	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A ( F |` ( C X. D ) ) B ) = ( A F B ) )
2	1	adantl 277	. 2 |- ( ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) /\ ( A e. C /\ B e. D ) ) -> ( A ( F |` ( C X. D ) ) B ) = ( A F B ) )
3		fndm 5480	. . . . . . 7 |- ( G Fn ( C X. D ) -> dom G = ( C X. D ) )
4	3	reseq2d 5063	. . . . . 6 |- ( G Fn ( C X. D ) -> ( F |` dom G ) = ( F |` ( C X. D ) ) )
5	4	3ad2ant2 1050	. . . . 5 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` dom G ) = ( F |` ( C X. D ) ) )
6		funssres 5420	. . . . . 6 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
7	6	3adant2 1047	. . . . 5 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` dom G ) = G )
8	5, 7	eqtr3d 2273	. . . 4 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` ( C X. D ) ) = G )
9	8	oveqd 6102	. . 3 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( A ( F |` ( C X. D ) ) B ) = ( A G B ) )
10	9	adantr 276	. 2 |- ( ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) /\ ( A e. C /\ B e. D ) ) -> ( A ( F |` ( C X. D ) ) B ) = ( A G B ) )
11	2, 10	eqtr3d 2273	1 |- ( ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) /\ ( A e. C /\ B e. D ) ) -> ( A F B ) = ( A G B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   X. cxp 4772   dom cdm 4774   |` cres 4776   Fun wfun 5371   Fn wfn 5372  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088

This theorem is used by: (None)