Theorem oprssov 6065

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 6063	. . 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 5357	. . . . . . 7 |- ( G Fn ( C X. D ) -> dom G = ( C X. D ) )
4	3	reseq2d 4946	. . . . . 6 |- ( G Fn ( C X. D ) -> ( F |` dom G ) = ( F |` ( C X. D ) ) )
5	4	3ad2ant2 1021	. . . . 5 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` dom G ) = ( F |` ( C X. D ) ) )
6		funssres 5300	. . . . . 6 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
7	6	3adant2 1018	. . . . 5 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` dom G ) = G )
8	5, 7	eqtr3d 2231	. . . 4 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` ( C X. D ) ) = G )
9	8	oveqd 5939	. . 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 2231	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 980   = wceq 1364   e. wcel 2167   C_ wss 3157   X. cxp 4661   dom cdm 4663   |` cres 4665   Fun wfun 5252   Fn wfn 5253  (class class class)co 5922

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925

This theorem is referenced by: (None)