ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ovmpt4g Unicode version

Theorem ovmpt4g 5861
Description: Value of a function given by the maps-to notation. (This is the operation analog of fvmpt2 5472.) (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 2674 . . 3  |-  ( C  e.  V  ->  E. z 
z  =  C )
2 moeq 2832 . . . . . . 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-mpo 5747 . . . . . . 7  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
64, 5eqtri 2138 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
73, 6ovidi 5857 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( z  =  C  ->  ( x F y )  =  z ) )
8 eqeq2 2127 . . . . 5  |-  ( z  =  C  ->  (
( x F y )  =  z  <->  ( x F y )  =  C ) )
97, 8mpbidi 150 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( z  =  C  ->  ( x F y )  =  C ) )
109exlimdv 1775 . . 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 1163 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 103    /\ w3a 947    = wceq 1316   E.wex 1453    e. wcel 1465   E*wmo 1978  (class class class)co 5742   {coprab 5743    e. cmpo 5744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747
This theorem is referenced by:  ovmpos  5862  ov2gf  5863  ovmpodxf  5864  ovmpodf  5870  ofmres  6002  fnmpoovd  6080  mapxpen  6710  cnmpt21  12387  cnmpt2t  12389  cnmptcom  12394
  Copyright terms: Public domain W3C validator