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Theorem ovmpt2ga 5977
Description: Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovmpt2ga.1  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
ovmpt2ga.2  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
Assertion
Ref Expression
ovmpt2ga  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  H )  ->  ( A F B )  =  S )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y   
x, S, y
Allowed substitution hints:    R( x, y)    F( x, y)    H( x, y)

Proof of Theorem ovmpt2ga
StepHypRef Expression
1 elex 2796 . 2  |-  ( S  e.  H  ->  S  e.  _V )
2 ovmpt2ga.2 . . . 4  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
32a1i 10 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
4 ovmpt2ga.1 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
54adantl 452 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
6 simp1 955 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  A  e.  C )
7 simp2 956 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  B  e.  D )
8 simp3 957 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  S  e.  _V )
93, 5, 6, 7, 8ovmpt2d 5975 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  ( A F B )  =  S )
101, 9syl3an3 1217 1  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  H )  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788  (class class class)co 5858    e. cmpt2 5860
This theorem is referenced by:  ovmpt2a  5978  ovmpt2g  5982  elovmpt2  6064  offval  6085  offval3  6091  hashbcval  13049  setsvalg  13171  ressval  13195  restval  13331  sylow1lem4  14912  sylow3lem2  14939  sylow3lem3  14940  lsmvalx  14950  mvrfval  16165  opsrval  16216  cnmpt12  17361  cnmpt22  17368  qtopval  17386  flimval  17658  fclsval  17703  stdbdmetval  18060  ofcfval3  23463  isdivcv2  25693  isder  25707  issrc  25867  isntr  25873  islimcat  25876  indcls2  25996  lineval222  26079  lineval3a  26083  fmulcl  27711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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