Theorem oprssov 6196

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 6194	. . 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 5455	. . . . . . 7 |- ( G Fn ( C X. D ) -> dom G = ( C X. D ) )
4	3	reseq2d 5038	. . . . . 6 |- ( G Fn ( C X. D ) -> ( F |` dom G ) = ( F |` ( C X. D ) ) )
5	4	3ad2ant2 1046	. . . . 5 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` dom G ) = ( F |` ( C X. D ) ) )
6		funssres 5395	. . . . . 6 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
7	6	3adant2 1043	. . . . 5 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` dom G ) = G )
8	5, 7	eqtr3d 2267	. . . 4 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` ( C X. D ) ) = G )
9	8	oveqd 6067	. . 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 2267	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 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   C_ wss 3211   X. cxp 4747   dom cdm 4749   |` cres 4751   Fun wfun 5346   Fn wfn 5347  (class class class)co 6050

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ov 6053

This theorem is referenced by: (None)