Theorem mpomptsx 6255

Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
mpomptsx	|- ( x e. A , y e. B |-> C ) = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )

Distinct variable groups:   x, y, z, A   y, B, z   z, C

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

Proof of Theorem mpomptsx

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2766	. . . . . 6 |- u e. _V
2		vex 2766	. . . . . 6 |- v e. _V
3	1, 2	op1std 6206	. . . . 5 |- ( z = <. u , v >. -> ( 1st ` z ) = u )
4	3	csbeq1d 3091	. . . 4 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C )
5	1, 2	op2ndd 6207	. . . . . 6 |- ( z = <. u , v >. -> ( 2nd ` z ) = v )
6	5	csbeq1d 3091	. . . . 5 |- ( z = <. u , v >. -> [_ ( 2nd ` z ) / y ]_ C = [_ v / y ]_ C )
7	6	csbeq2dv 3110	. . . 4 |- ( z = <. u , v >. -> [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
8	4, 7	eqtrd 2229	. . 3 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
9	8	mpomptx 6013	. 2 |- ( z e. U_ u e. A ( { u } X. [_ u / x ]_ B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C ) = ( u e. A , v e. [_ u / x ]_ B |-> [_ u / x ]_ [_ v / y ]_ C )
10		nfcv 2339	. . . 4 |- F/_ u ( { x } X. B )
11		nfcv 2339	. . . . 5 |- F/_ x { u }
12		nfcsb1v 3117	. . . . 5 |- F/_ x [_ u / x ]_ B
13	11, 12	nfxp 4690	. . . 4 |- F/_ x ( { u } X. [_ u / x ]_ B )
14		sneq 3633	. . . . 5 |- ( x = u -> { x } = { u } )
15		csbeq1a 3093	. . . . 5 |- ( x = u -> B = [_ u / x ]_ B )
16	14, 15	xpeq12d 4688	. . . 4 |- ( x = u -> ( { x } X. B ) = ( { u } X. [_ u / x ]_ B ) )
17	10, 13, 16	cbviun 3953	. . 3 |- U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B )
18		mpteq1 4117	. . 3 |- ( U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B ) -> ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C ) = ( z e. U_ u e. A ( { u } X. [_ u / x ]_ B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C ) )
19	17, 18	ax-mp 5	. 2 |- ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C ) = ( z e. U_ u e. A ( { u } X. [_ u / x ]_ B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
20		nfcv 2339	. . 3 |- F/_ u B
21		nfcv 2339	. . 3 |- F/_ u C
22		nfcv 2339	. . 3 |- F/_ v C
23		nfcsb1v 3117	. . 3 |- F/_ x [_ u / x ]_ [_ v / y ]_ C
24		nfcv 2339	. . . 4 |- F/_ y u
25		nfcsb1v 3117	. . . 4 |- F/_ y [_ v / y ]_ C
26	24, 25	nfcsb 3122	. . 3 |- F/_ y [_ u / x ]_ [_ v / y ]_ C
27		csbeq1a 3093	. . . 4 |- ( y = v -> C = [_ v / y ]_ C )
28		csbeq1a 3093	. . . 4 |- ( x = u -> [_ v / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
29	27, 28	sylan9eqr 2251	. . 3 |- ( ( x = u /\ y = v ) -> C = [_ u / x ]_ [_ v / y ]_ C )
30	20, 12, 21, 22, 23, 26, 15, 29	cbvmpox 6000	. 2 |- ( x e. A , y e. B |-> C ) = ( u e. A , v e. [_ u / x ]_ B |-> [_ u / x ]_ [_ v / y ]_ C )
31	9, 19, 30	3eqtr4ri 2228	1 |- ( x e. A , y e. B |-> C ) = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )

Syntax hints:   = wceq 1364   [_csb 3084   {csn 3622   <.cop 3625   U_ciun 3916   |-> cmpt 4094   X. cxp 4661   ` cfv 5258   e. cmpo 5924   1stc1st 6196   2ndc2nd 6197

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fv 5266  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199

This theorem is referenced by:  mpompts  6256  mpofvex  6263