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

Step	Hyp	Ref	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 ) )
3	1, 2	ax-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 )
5	4	mpt2mptx 5731	. 2 |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) = ( x e. A , y e. B |-> D )
6	3, 5	eqtr3i 2110	1 |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> D )

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