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Theorem ovmpt4g 6188
Description: Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5804.) (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 2958 . . 3  |-  ( C  e.  V  ->  E. z 
z  =  C )
2 moeq 3102 . . . . . . 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 6078 . . . . . . 7  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
64, 5eqtri 2455 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
73, 6ovidi 6184 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( z  =  C  ->  ( x F y )  =  z ) )
8 eqeq2 2444 . . . . 5  |-  ( z  =  C  ->  (
( x F y )  =  z  <->  ( x F y )  =  C ) )
97, 8mpbidi 208 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( z  =  C  ->  ( x F y )  =  C ) )
109exlimdv 1646 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( E. z  z  =  C  ->  (
x F y )  =  C ) )
111, 10syl5 30 . 2  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( C  e.  V  ->  ( x F y )  =  C ) )
12113impia 1150 1  |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  ->  ( x F y )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   E*wmo 2281  (class class class)co 6073   {coprab 6074    e. cmpt2 6075
This theorem is referenced by:  ovmpt2s  6189  ov2gf  6190  ovmpt2dxf  6191  ovmpt2df  6197  ofmres  6335  mapxpen  7265  pwfseqlem2  8526  pwfseqlem3  8527  fullfunc  14095  fthfunc  14096  prfcl  14292  curf1cl  14317  curfcl  14321  hofcl  14348  gsum2d2lem  15539  gsum2d2  15540  gsumcom2  15541  dprdval  15553  dprd2d2  15594  cnmpt21  17695  cnmpt2t  17697  cnmptcom  17702  cnmpt2k  17712  xkocnv  17838  sdclem2  26437  aovmpt4g  28032
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078
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