Theorem mpomptx 6009

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

Assertion

Ref	Expression
mpomptx	|- ( 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 mpomptx

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 4092	. 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-mpo 5923	. . 3 |- ( x e. A , y e. B |-> D ) = { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = D ) }
3		eliunxp 4801	. . . . . . 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 458	. . . . . 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 1915	. . . . . 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 401	. . . . . . . 8 |- ( ( ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) /\ w = C ) <-> ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = C ) ) )
7		mpompt.1	. . . . . . . . . . 11 |- ( z = <. x , y >. -> C = D )
8	7	eqeq2d 2205	. . . . . . . . . 10 |- ( z = <. x , y >. -> ( w = C <-> w = D ) )
9	8	anbi2d 464	. . . . . . . . 9 |- ( z = <. x , y >. -> ( ( ( x e. A /\ y e. B ) /\ w = C ) <-> ( ( x e. A /\ y e. B ) /\ w = D ) ) )
10	9	pm5.32i 454	. . . . . . . 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 184	. . . . . . 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 1617	. . . . . 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 208	. . . . 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 4096	. . . 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 5965	. . . 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 2217	. . 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 2217	. 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 2217	1 |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2164   {csn 3618   <.cop 3621   U_ciun 3912   {copab 4089   |-> cmpt 4090   X. cxp 4657   {coprab 5919   e. cmpo 5920

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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-iun 3914  df-opab 4091  df-mpt 4092  df-xp 4665  df-rel 4666  df-oprab 5922  df-mpo 5923

This theorem is referenced by:  mpompt  6010  mpomptsx  6250  dmmpossx  6252  fmpox  6253