Theorem oprssov 5662

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 5660	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A ( F |` ( C X. D ) ) B ) = ( A F B ) )
2	1	adantl 271	. 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 5018	. . . . . . 7 |- ( G Fn ( C X. D ) -> dom G = ( C X. D ) )
4	3	reseq2d 4630	. . . . . 6 |- ( G Fn ( C X. D ) -> ( F |` dom G ) = ( F |` ( C X. D ) ) )
5	4	3ad2ant2 960	. . . . 5 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` dom G ) = ( F |` ( C X. D ) ) )
6		funssres 4962	. . . . . 6 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
7	6	3adant2 957	. . . . 5 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` dom G ) = G )
8	5, 7	eqtr3d 2115	. . . 4 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` ( C X. D ) ) = G )
9	8	oveqd 5549	. . 3 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( A ( F |` ( C X. D ) ) B ) = ( A G B ) )
10	9	adantr 270	. 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 2115	1 |- ( ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) /\ ( A e. C /\ B e. D ) ) -> ( A F B ) = ( A G B ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   C_ wss 2973   X. cxp 4361   dom cdm 4363   |` cres 4365   Fun wfun 4916   Fn wfn 4917  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930  df-ov 5535

This theorem is referenced by: (None)