Theorem oprssov 6015

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 6013	. . 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 5315	. . . . . . 7 |- ( G Fn ( C X. D ) -> dom G = ( C X. D ) )
4	3	reseq2d 4907	. . . . . 6 |- ( G Fn ( C X. D ) -> ( F |` dom G ) = ( F |` ( C X. D ) ) )
5	4	3ad2ant2 1019	. . . . 5 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` dom G ) = ( F |` ( C X. D ) ) )
6		funssres 5258	. . . . . 6 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
7	6	3adant2 1016	. . . . 5 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` dom G ) = G )
8	5, 7	eqtr3d 2212	. . . 4 |- ( ( Fun F /\ G Fn ( C X. D ) /\ G C_ F ) -> ( F |` ( C X. D ) ) = G )
9	8	oveqd 5891	. . 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 2212	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 978   = wceq 1353   e. wcel 2148   C_ wss 3129   X. cxp 4624   dom cdm 4626   |` cres 4628   Fun wfun 5210   Fn wfn 5211  (class class class)co 5874

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877

This theorem is referenced by: (None)