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Theorem mpt2mpt 6291
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 5002 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
2 mpteq1 4479 . . 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 mpt2mpt.1 . . 3  |-  ( z  =  <. x ,  y
>.  ->  C  =  D )
54mpt2mptx 6290 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
63, 5eqtr3i 2485 1  |-  ( z  e.  ( A  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   {csn 3984   <.cop 3990   U_ciun 4278    |-> cmpt 4457    X. cxp 4945    |-> cmpt2 6201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-iun 4280  df-opab 4458  df-mpt 4459  df-xp 4953  df-rel 4954  df-oprab 6203  df-mpt2 6204
This theorem is referenced by:  fconstmpt2  6294  fnov  6307  fmpt2co  6765  xpf1o  7582  resfval2  14921  catcisolem  15092  xpccatid  15116  curf2ndf  15175  evlslem4OLD  17713  evlslem4  17714  mdetunilem9  18557  txbas  19271  cnmpt1st  19372  cnmpt2nd  19373  cnmpt2c  19374  cnmpt2t  19377  txhmeo  19507  txswaphmeolem  19508  ptuncnv  19511  ptunhmeo  19512  xpstopnlem1  19513  xkohmeo  19519  prdstmdd  19825  ucnimalem  19986  fmucndlem  19997  fsum2cn  20578  lmod1zr  31153
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