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Theorem mpt2mpt 5732
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 4494 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
2 mpteq1 3920 . . 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 7 . 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 5731 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
63, 5eqtr3i 2110 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 1289   {csn 3444   <.cop 3447   U_ciun 3728    |-> cmpt 3897    X. cxp 4434    |-> cmpt2 5646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-iun 3730  df-opab 3898  df-mpt 3899  df-xp 4442  df-rel 4443  df-oprab 5648  df-mpt2 5649
This theorem is referenced by:  fnovim  5745  fmpt2co  5973  xpf1o  6550
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