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Theorem ovmpt4g 6212
Description: Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5778.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)
Hypothesis
Ref Expression
ovmpt4g.3  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
ovmpt4g  |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  ->  ( x F y )  =  C )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)    F( x, y)    V( x, y)

Proof of Theorem ovmpt4g
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elisset 2981 . . 3  |-  ( C  e.  V  ->  E. z 
z  =  C )
2 moeq 3132 . . . . . . 7  |-  E* z 
z  =  C
32a1i 11 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E* z  z  =  C )
4 ovmpt4g.3 . . . . . . 7  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
5 df-mpt2 6095 . . . . . . 7  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
64, 5eqtri 2461 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
73, 6ovidi 6208 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( z  =  C  ->  ( x F y )  =  z ) )
8 eqeq2 2450 . . . . 5  |-  ( z  =  C  ->  (
( x F y )  =  z  <->  ( x F y )  =  C ) )
97, 8mpbidi 216 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( z  =  C  ->  ( x F y )  =  C ) )
109exlimdv 1695 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( E. z  z  =  C  ->  (
x F y )  =  C ) )
111, 10syl5 32 . 2  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( C  e.  V  ->  ( x F y )  =  C ) )
12113impia 1179 1  |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  ->  ( x F y )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364   E.wex 1591    e. wcel 1761   E*wmo 2260  (class class class)co 6090   {coprab 6091    e. cmpt2 6092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095
This theorem is referenced by:  ovmpt2s  6213  ov2gf  6214  ovmpt2dxf  6215  ovmpt2df  6221  ofmres  6572  fnmpt2ovd  6650  mapxpen  7473  pwfseqlem2  8822  pwfseqlem3  8823  fullfunc  14812  fthfunc  14813  prfcl  15009  curf1cl  15034  curfcl  15038  hofcl  15065  gsum2d2lem  16455  gsum2d2  16456  gsumcom2  16457  dprdval  16475  dprdvalOLD  16477  dprd2d2  16533  cnmpt21  19144  cnmpt2t  19146  cnmptcom  19151  cnmpt2k  19161  xkocnv  19287  sdclem2  28547  aovmpt4g  30016  ovmpt2rdxf  30637
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