Theorem mpt2mptx 5615

Description: Express a two-argument function as a one-argument function, or vice-versa. In this version B ( x ) is not assumed to be constant w.r.t x. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypothesis

Ref	Expression
mpt2mpt.1	|- ( z = <. x , y >. -> C = D )

Assertion

Ref	Expression
mpt2mptx	|- ( z e. U_ x e. A ( { x } 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

Allowed substitution hints:   B( x)   C( z)   D( x, y)

Proof of Theorem mpt2mptx

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 3841	. 2 |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) = { <. z , w >. | ( z e. U_ x e. A ( { x } X. B ) /\ w = C ) }
2		df-mpt2 5537	. . 3 |- ( x e. A , y e. B |-> D ) = { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = D ) }
3		eliunxp 4493	. . . . . . 7 |- ( z e. U_ x e. A ( { x } X. B ) <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) )
4	3	anbi1i 445	. . . . . 6 |- ( ( z e. U_ x e. A ( { x } X. B ) /\ w = C ) <-> ( E. x E. y ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) /\ w = C ) )
5		19.41vv 1824	. . . . . 6 |- ( E. x E. y ( ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) /\ w = C ) <-> ( E. x E. y ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) /\ w = C ) )
6		anass 393	. . . . . . . 8 |- ( ( ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) /\ w = C ) <-> ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = C ) ) )
7		mpt2mpt.1	. . . . . . . . . . 11 |- ( z = <. x , y >. -> C = D )
8	7	eqeq2d 2092	. . . . . . . . . 10 |- ( z = <. x , y >. -> ( w = C <-> w = D ) )
9	8	anbi2d 451	. . . . . . . . 9 |- ( z = <. x , y >. -> ( ( ( x e. A /\ y e. B ) /\ w = C ) <-> ( ( x e. A /\ y e. B ) /\ w = D ) ) )
10	9	pm5.32i 441	. . . . . . . 8 |- ( ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = C ) ) <-> ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = D ) ) )
11	6, 10	bitri 182	. . . . . . 7 |- ( ( ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) /\ w = C ) <-> ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = D ) ) )
12	11	2exbii 1537	. . . . . 6 |- ( E. x E. y ( ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) /\ w = C ) <-> E. x E. y ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = D ) ) )
13	4, 5, 12	3bitr2i 206	. . . . 5 |- ( ( z e. U_ x e. A ( { x } X. B ) /\ w = C ) <-> E. x E. y ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = D ) ) )
14	13	opabbii 3845	. . . 4 |- { <. z , w >. | ( z e. U_ x e. A ( { x } X. B ) /\ w = C ) } = { <. z , w >. | E. x E. y ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = D ) ) }
15		dfoprab2 5572	. . . 4 |- { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = D ) } = { <. z , w >. | E. x E. y ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = D ) ) }
16	14, 15	eqtr4i 2104	. . 3 |- { <. z , w >. | ( z e. U_ x e. A ( { x } X. B ) /\ w = C ) } = { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = D ) }
17	2, 16	eqtr4i 2104	. 2 |- ( x e. A , y e. B |-> D ) = { <. z , w >. | ( z e. U_ x e. A ( { x } X. B ) /\ w = C ) }
18	1, 17	eqtr4i 2104	1 |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   {csn 3398   <.cop 3401   U_ciun 3678   {copab 3838   |-> cmpt 3839   X. cxp 4361   {coprab 5533   |-> cmpt2 5534

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-iun 3680  df-opab 3840  df-mpt 3841  df-xp 4369  df-rel 4370  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  mpt2mpt  5616  mpt2mptsx  5843  dmmpt2ssx  5845  fmpt2x  5846