Theorem mpomptx 6152

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 4178	. 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 6063	. . 3 |- ( x e. A , y e. B |-> D ) = { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = D ) }
3		eliunxp 4899	. . . . . . 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 1955	. . . . . 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 2246	. . . . . . . . . 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 1655	. . . . . 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 4182	. . . 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 6108	. . . 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 2258	. . 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 2258	. 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 2258	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 1398   E.wex 1541   e. wcel 2205   {csn 3694   <.cop 3697   U_ciun 3996   {copab 4175   |-> cmpt 4176   X. cxp 4752   {coprab 6059   e. cmpo 6060

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-iun 3998  df-opab 4177  df-mpt 4178  df-xp 4760  df-rel 4761  df-oprab 6062  df-mpo 6063

This theorem is referenced by:  mpompt  6153  mpomptsx  6406  dmmpossx  6408  fmpox  6409