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Theorem ovmpt2ga 6333
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 3087 . 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 466 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
6 simp1 988 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  A  e.  C )
7 simp2 989 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  B  e.  D )
8 simp3 990 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  S  e.  _V )
93, 5, 6, 7, 8ovmpt2d 6331 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  ( A F B )  =  S )
101, 9syl3an3 1254 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078  (class class class)co 6203    |-> cmpt2 6205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208
This theorem is referenced by:  ovmpt2a  6334  ovmpt2g  6338  elovmpt2  6420  offval  6440  offval3  6684  bropopvvv  6766  reps  12529  hashbcval  14184  setsvalg  14318  ressval  14347  restval  14487  sylow1lem4  16224  sylow3lem2  16251  sylow3lem3  16252  lsmvalx  16262  mvrfval  17620  opsrval  17683  marrepfval  18501  marrepval0  18502  marepvfval  18506  marepvval0  18507  cnmpt12  19375  cnmpt22  19382  qtopval  19403  flimval  19671  fclsval  19716  ucnval  19987  stdbdmetval  20224  wlkon  23601  trlon  23611  pthon  23646  spthon  23653  resvval  26460  ofcfval3  26709  fmulcl  29930  is2wlkonot  30550  is2spthonot  30551  2wlkonot  30552  2spthonot  30553  2wlksot  30554  2spthsot  30555  2wlkonot3v  30562  2spthonot3v  30563
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