MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oprssov Structured version   Unicode version

Theorem oprssov 6227
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
StepHypRef Expression
1 ovres 6225 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A ( F  |`  ( C  X.  D
) ) B )  =  ( A F B ) )
21adantl 466 . 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 5505 . . . . . . 7  |-  ( G  Fn  ( C  X.  D )  ->  dom  G  =  ( C  X.  D ) )
43reseq2d 5105 . . . . . 6  |-  ( G  Fn  ( C  X.  D )  ->  ( F  |`  dom  G )  =  ( F  |`  ( C  X.  D
) ) )
543ad2ant2 1010 . . . . 5  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |` 
dom  G )  =  ( F  |`  ( C  X.  D ) ) )
6 funssres 5453 . . . . . 6  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
763adant2 1007 . . . . 5  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |` 
dom  G )  =  G )
85, 7eqtr3d 2472 . . . 4  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |`  ( C  X.  D
) )  =  G )
98oveqd 6103 . . 3  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( A
( F  |`  ( C  X.  D ) ) B )  =  ( A G B ) )
109adantr 465 . 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 ) )
112, 10eqtr3d 2472 1  |-  ( ( ( Fun  F  /\  G  Fn  ( C  X.  D )  /\  G  C_  F )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  ( A F B )  =  ( A G B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3323    X. cxp 4833   dom cdm 4835    |` cres 4837   Fun wfun 5407    Fn wfn 5408  (class class class)co 6086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-res 4847  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421  df-ov 6089
This theorem is referenced by:  sspg  24077  ssps  24079
  Copyright terms: Public domain W3C validator