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Theorem mpt2mpt 5939
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
mpt2mpt.1  |-  ( z  =  <. x ,  y
>.  ->  C  =  D )
Assertion
Ref Expression
mpt2mpt  |-  ( 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 mpt2mpt
StepHypRef Expression
1 iunxpconst 4746 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
2 mpteq1 4100 . . 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 8 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( z  e.  ( A  X.  B )  |->  C )
4 mpt2mpt.1 . . 3  |-  ( z  =  <. x ,  y
>.  ->  C  =  D )
54mpt2mptx 5938 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
63, 5eqtr3i 2305 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 1623   {csn 3640   <.cop 3643   U_ciun 3905    e. cmpt 4077    X. cxp 4687    e. cmpt2 5860
This theorem is referenced by:  fnov  5952  fmpt2co  6202  xpf1o  7023  resfval2  13767  catcisolem  13938  xpccatid  13962  curf2ndf  14021  evlslem4  16245  txbas  17262  cnmpt1st  17362  cnmpt2nd  17363  cnmpt2c  17364  cnmpt2t  17367  txhmeo  17494  txswaphmeolem  17495  ptuncnv  17498  ptunhmeo  17499  xpstopnlem1  17500  xkohmeo  17506  prdstmdd  17806  fsum2cn  18375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iun 3907  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-oprab 5862  df-mpt2 5863
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