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Theorem ovmpt4g 6408
Description: Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5956.) (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 3124 . . 3  |-  ( C  e.  V  ->  E. z 
z  =  C )
2 moeq 3279 . . . . . . 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 6288 . . . . . . 7  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
64, 5eqtri 2496 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
73, 6ovidi 6404 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( z  =  C  ->  ( x F y )  =  z ) )
8 eqeq2 2482 . . . . 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 1700 . . 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 1193 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 973    = wceq 1379   E.wex 1596    e. wcel 1767   E*wmo 2276  (class class class)co 6283   {coprab 6284    |-> cmpt2 6285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5550  df-fun 5589  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288
This theorem is referenced by:  ovmpt2s  6409  ov2gf  6410  ovmpt2dxf  6411  ovmpt2df  6417  ofmres  6780  fnmpt2ovd  6861  mapxpen  7683  pwfseqlem2  9036  pwfseqlem3  9037  fullfunc  15132  fthfunc  15133  prfcl  15329  curf1cl  15354  curfcl  15358  hofcl  15385  gsum2d2lem  16801  gsum2d2  16802  gsumcom2  16803  dprdval  16834  dprdvalOLD  16836  dprd2d2  16892  cnmpt21  19923  cnmpt2t  19925  cnmptcom  19930  cnmpt2k  19940  xkocnv  20066  sdclem2  29854  aovmpt4g  31769  ovmpt2rdxf  32012
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