Theorem mpt2mptxf 29477

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.) (Revised by Thierry Arnoux, 31-Mar-2018.)

Hypotheses

Ref	Expression
mpt2mptxf.0	|- F/_ x C
mpt2mptxf.1	|- F/_ y C
mpt2mptxf.2	|- ( z = <. x , y >. -> C = D )

Assertion

Ref	Expression
mpt2mptxf	|- ( 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   z, D

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

Proof of Theorem mpt2mptxf

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 4730	. 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 6655	. . 3 |- ( x e. A , y e. B |-> D ) = { <. <. x , y >. , w >. | ( ( x e. A /\ y e. B ) /\ w = D ) }
3		eliunxp 5259	. . . . . . 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 731	. . . . . 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		mpt2mptxf.1	. . . . . . . . . 10 |- F/_ y C
6	5	nfeq2 2780	. . . . . . . . 9 |- F/ y w = C
7	6	19.41 2103	. . . . . . . 8 |- ( E. y ( ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) /\ w = C ) <-> ( E. y ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) /\ w = C ) )
8	7	exbii 1774	. . . . . . 7 |- ( 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 ) )
9		mpt2mptxf.0	. . . . . . . . 9 |- F/_ x C
10	9	nfeq2 2780	. . . . . . . 8 |- F/ x w = C
11	10	19.41 2103	. . . . . . 7 |- ( 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 ) )
12	8, 11	bitri 264	. . . . . 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 ) )
13		anass 681	. . . . . . . 8 |- ( ( ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) /\ w = C ) <-> ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = C ) ) )
14		mpt2mptxf.2	. . . . . . . . . . 11 |- ( z = <. x , y >. -> C = D )
15	14	eqeq2d 2632	. . . . . . . . . 10 |- ( z = <. x , y >. -> ( w = C <-> w = D ) )
16	15	anbi2d 740	. . . . . . . . 9 |- ( z = <. x , y >. -> ( ( ( x e. A /\ y e. B ) /\ w = C ) <-> ( ( x e. A /\ y e. B ) /\ w = D ) ) )
17	16	pm5.32i 669	. . . . . . . 8 |- ( ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = C ) ) <-> ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = D ) ) )
18	13, 17	bitri 264	. . . . . . 7 |- ( ( ( z = <. x , y >. /\ ( x e. A /\ y e. B ) ) /\ w = C ) <-> ( z = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ w = D ) ) )
19	18	2exbii 1775	. . . . . 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 ) ) )
20	4, 12, 19	3bitr2i 288	. . . . 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 ) ) )
21	20	opabbii 4717	. . . 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 ) ) }
22		dfoprab2 6701	. . . 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 ) ) }
23	21, 22	eqtr4i 2647	. . 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 ) }
24	2, 23	eqtr4i 2647	. 2 |- ( x e. A , y e. B |-> D ) = { <. z , w >. | ( z e. U_ x e. A ( { x } X. B ) /\ w = C ) }
25	1, 24	eqtr4i 2647	1 |- ( z e. U_ x e. A ( { x } X. B ) |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   F/_wnfc 2751   {csn 4177   <.cop 4183   U_ciun 4520   {copab 4712   |-> cmpt 4729   X. cxp 5112   {coprab 6651   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  gsummpt2co  29780