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Theorem mpt2mpt 6157
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 4926 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
2 mpteq1 4281 . . 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 6156 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
63, 5eqtr3i 2457 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 1652   {csn 3806   <.cop 3809   U_ciun 4085    e. cmpt 4258    X. cxp 4868    e. cmpt2 6075
This theorem is referenced by:  fnov  6170  fmpt2co  6422  xpf1o  7261  resfval2  14082  catcisolem  14253  xpccatid  14277  curf2ndf  14336  evlslem4  16556  txbas  17591  cnmpt1st  17692  cnmpt2nd  17693  cnmpt2c  17694  cnmpt2t  17697  txhmeo  17827  txswaphmeolem  17828  ptuncnv  17831  ptunhmeo  17832  xpstopnlem1  17833  xkohmeo  17839  prdstmdd  18145  ucnimalem  18302  fmucndlem  18313  fsum2cn  18893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-iun 4087  df-opab 4259  df-mpt 4260  df-xp 4876  df-rel 4877  df-oprab 6077  df-mpt2 6078
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