Theorem mpomptx 6111

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 4152	. 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 6022	. . 3 |- ( x e. A , y e. B |-> D ) = { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = D ) }
3		eliunxp 4869	. . . . . . 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 1952	. . . . . 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 2243	. . . . . . . . . 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 1654	. . . . . 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 4156	. . . 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 6067	. . . 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 2255	. . 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 2255	. 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 2255	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 1397   E.wex 1540   e. wcel 2202   {csn 3669   <.cop 3672   U_ciun 3970   {copab 4149   |-> cmpt 4150   X. cxp 4723   {coprab 6018   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:  mpompt  6112  mpomptsx  6361  dmmpossx  6363  fmpox  6364