Theorem mpompt 6112

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
mpompt.1	|- ( z = <. x , y >. -> C = D )

Assertion

Ref	Expression
mpompt	|- ( 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 mpompt

Step	Hyp	Ref	Expression
1		iunxpconst 4786	. . 3 |- U_ x e. A ( { x } X. B ) = ( A X. B )
2		mpteq1 4173	. . 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		mpompt.1	. . 3 |- ( z = <. x , y >. -> C = D )
5	4	mpomptx 6111	. 2 |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) = ( x e. A , y e. B |-> D )
6	3, 5	eqtr3i 2254	1 |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   = wceq 1397   {csn 3669   <.cop 3672   U_ciun 3970   |-> cmpt 4150   X. cxp 4723   e. cmpo 6019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-iun 3972  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-oprab 6021  df-mpo 6022

This theorem is referenced by:  fconstmpo  6115  fnovim  6129  fmpoco  6380  xpf1o  7029  txbas  14981  cnmpt1st  15011  cnmpt2nd  15012  cnmpt2c  15013  cnmpt2t  15016  txhmeo  15042  txswaphmeolem  15043