Theorem mpompt 6145

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 4810	. . 3 |- U_ x e. A ( { x } X. B ) = ( A X. B )
2		mpteq1 4194	. . 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 6144	. 2 |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) = ( x e. A , y e. B |-> D )
6	3, 5	eqtr3i 2255	1 |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   = wceq 1398   {csn 3689   <.cop 3692   U_ciun 3991   |-> cmpt 4171   X. cxp 4747   e. cmpo 6052

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-iun 3993  df-opab 4172  df-mpt 4173  df-xp 4755  df-rel 4756  df-oprab 6054  df-mpo 6055

This theorem is referenced by:  fconstmpo  6148  fnovim  6162  fmpoco  6412  xpf1o  7097  txbas  15123  cnmpt1st  15153  cnmpt2nd  15154  cnmpt2c  15155  cnmpt2t  15158  txhmeo  15184  txswaphmeolem  15185