Theorem mpt2mpt 6752

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 5175	. . 3 |- U_ x e. A ( { x } X. B ) = ( A X. B )
2		mpteq1 4737	. . 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 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 )
5	4	mpt2mptx 6751	. 2 |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) = ( x e. A , y e. B |-> D )
6	3, 5	eqtr3i 2646	1 |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   = wceq 1483   {csn 4177   <.cop 4183   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  fconstmpt2  6755  fnov  6768  fmpt2co  7260  xpf1o  8122  resfval2  16553  catcisolem  16756  xpccatid  16828  curf2ndf  16887  evlslem4  19508  mdetunilem9  20426  txbas  21370  cnmpt1st  21471  cnmpt2nd  21472  cnmpt2c  21473  cnmpt2t  21476  txhmeo  21606  txswaphmeolem  21607  ptuncnv  21610  ptunhmeo  21611  xpstopnlem1  21612  xkohmeo  21618  prdstmdd  21927  ucnimalem  22084  fmucndlem  22095  fsum2cn  22674  fimaproj  29900  curfv  33389  idfusubc0  41865  lmod1zr  42282