MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovmpt2ga Structured version   Unicode version

Theorem ovmpt2ga 6331
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 3043 . 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 464 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
6 simp1 994 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  A  e.  C )
7 simp2 995 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  B  e.  D )
8 simp3 996 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  S  e.  _V )
93, 5, 6, 7, 8ovmpt2d 6329 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  ( A F B )  =  S )
101, 9syl3an3 1261 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   _Vcvv 3034  (class class class)co 6196    |-> cmpt2 6198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201
This theorem is referenced by:  ovmpt2a  6332  ovmpt2g  6336  elovmpt2  6419  offval  6446  offval3  6693  bropopvvv  6779  reps  12653  hashbcval  14522  setsvalg  14658  ressval  14688  restval  14834  sylow1lem4  16738  sylow3lem2  16765  sylow3lem3  16766  lsmvalx  16776  mvrfval  18189  opsrval  18252  marrepfval  19147  marrepval0  19148  marepvfval  19152  marepvval0  19153  cnmpt12  20253  cnmpt22  20260  qtopval  20281  flimval  20549  fclsval  20594  ucnval  20865  stdbdmetval  21102  wlkon  24654  trlon  24663  pthon  24698  spthon  24705  is2wlkonot  24984  is2spthonot  24985  2wlkonot  24986  2spthonot  24987  2wlksot  24988  2spthsot  24989  2wlkonot3v  24996  2spthonot3v  24997  resvval  27971  ofcfval3  28250  fmulcl  31741
  Copyright terms: Public domain W3C validator