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Theorem ovmpt4g 5651
Description: Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5282.) (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 2585 . . 3  |-  ( C  e.  V  ->  E. z 
z  =  C )
2 moeq 2739 . . . . . . 7  |-  E* z 
z  =  C
32a1i 9 . . . . . 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 5545 . . . . . . 7  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
64, 5eqtri 2076 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
73, 6ovidi 5647 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( z  =  C  ->  ( x F y )  =  z ) )
8 eqeq2 2065 . . . . 5  |-  ( z  =  C  ->  (
( x F y )  =  z  <->  ( x F y )  =  C ) )
97, 8mpbidi 144 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( z  =  C  ->  ( x F y )  =  C ) )
109exlimdv 1716 . . 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 1112 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 101    /\ w3a 896    = wceq 1259   E.wex 1397    e. wcel 1409   E*wmo 1917  (class class class)co 5540   {coprab 5541    |-> cmpt2 5542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545
This theorem is referenced by:  ovmpt2s  5652  ov2gf  5653  ovmpt2dxf  5654  ovmpt2df  5660  ofmres  5791
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