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Theorem mpompt 5870
Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
mpompt.1  |-  ( z  =  <. x ,  y
>.  ->  C  =  D )
Assertion
Ref Expression
mpompt  |-  ( z  e.  ( A  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
Distinct variable groups:    x, y, z, A    y, B, z   
x, C, y    z, D    x, B
Allowed substitution hints:    C( z)    D( x, y)

Proof of Theorem mpompt
StepHypRef Expression
1 iunxpconst 4606 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
2 mpteq1 4019 . . 3  |-  ( U_ x  e.  A  ( { x }  X.  B )  =  ( A  X.  B )  ->  ( z  e. 
U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( z  e.  ( A  X.  B
)  |->  C ) )
31, 2ax-mp 5 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( z  e.  ( A  X.  B )  |->  C )
4 mpompt.1 . . 3  |-  ( z  =  <. x ,  y
>.  ->  C  =  D )
54mpomptx 5869 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
63, 5eqtr3i 2163 1  |-  ( z  e.  ( A  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332   {csn 3531   <.cop 3534   U_ciun 3820    |-> cmpt 3996    X. cxp 4544    e. cmpo 5783
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-iun 3822  df-opab 3997  df-mpt 3998  df-xp 4552  df-rel 4553  df-oprab 5785  df-mpo 5786
This theorem is referenced by:  fconstmpo  5873  fnovim  5886  fmpoco  6120  xpf1o  6745  txbas  12464  cnmpt1st  12494  cnmpt2nd  12495  cnmpt2c  12496  cnmpt2t  12499  txhmeo  12525  txswaphmeolem  12526
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