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Theorem ovmpt2ga 6438
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 3091 . 2  |-  ( S  e.  H  ->  S  e.  _V )
2 ovmpt2ga.2 . . . 4  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
32a1i 11 . . 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 468 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
6 simp1 1006 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  A  e.  C )
7 simp2 1007 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  B  e.  D )
8 simp3 1008 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  S  e.  _V )
93, 5, 6, 7, 8ovmpt2d 6436 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  ( A F B )  =  S )
101, 9syl3an3 1300 1  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  H )  ->  ( A F B )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   _Vcvv 3082  (class class class)co 6303    |-> cmpt2 6305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308
This theorem is referenced by:  ovmpt2a  6439  ovmpt2g  6443  elovmpt2  6526  offval  6550  offval3  6799  bropopvvv  6885  reps  12869  hashbcval  14947  setsvalg  15138  ressval  15169  restval  15318  sylow1lem4  17246  sylow3lem2  17273  sylow3lem3  17274  lsmvalx  17284  mvrfval  18637  opsrval  18691  marrepfval  19577  marrepval0  19578  marepvfval  19582  marepvval0  19583  cnmpt12  20674  cnmpt22  20681  qtopval  20702  flimval  20970  fclsval  21015  ucnval  21284  stdbdmetval  21521  wlkon  25253  trlon  25262  pthon  25297  spthon  25304  is2wlkonot  25583  is2spthonot  25584  2wlkonot  25585  2spthonot  25586  2wlksot  25587  2spthsot  25588  2wlkonot3v  25595  2spthonot3v  25596  resvval  28592  ofcfval3  28925  fmulcl  37523
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