Theorem mpompt 5983

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 4701	. . 3 |- U_ x e. A ( { x } X. B ) = ( A X. B )
2		mpteq1 4102	. . 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 5982	. 2 |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) = ( x e. A , y e. B |-> D )
6	3, 5	eqtr3i 2212	1 |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   = wceq 1364   {csn 3607   <.cop 3610   U_ciun 3901   |-> cmpt 4079   X. cxp 4639   e. cmpo 5893

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-iun 3903  df-opab 4080  df-mpt 4081  df-xp 4647  df-rel 4648  df-oprab 5895  df-mpo 5896

This theorem is referenced by:  fconstmpo  5986  fnovim  6000  fmpoco  6235  xpf1o  6862  txbas  14155  cnmpt1st  14185  cnmpt2nd  14186  cnmpt2c  14187  cnmpt2t  14190  txhmeo  14216  txswaphmeolem  14217