Theorem mpomptsx 6371

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 2806	. . . . . 6 |- u e. _V
2		vex 2806	. . . . . 6 |- v e. _V
3	1, 2	op1std 6320	. . . . 5 |- ( z = <. u , v >. -> ( 1st ` z ) = u )
4	3	csbeq1d 3135	. . . 4 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C )
5	1, 2	op2ndd 6321	. . . . . 6 |- ( z = <. u , v >. -> ( 2nd ` z ) = v )
6	5	csbeq1d 3135	. . . . 5 |- ( z = <. u , v >. -> [_ ( 2nd ` z ) / y ]_ C = [_ v / y ]_ C )
7	6	csbeq2dv 3154	. . . 4 |- ( z = <. u , v >. -> [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
8	4, 7	eqtrd 2264	. . 3 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
9	8	mpomptx 6122	. 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 2375	. . . 4 |- F/_ u ( { x } X. B )
11		nfcv 2375	. . . . 5 |- F/_ x { u }
12		nfcsb1v 3161	. . . . 5 |- F/_ x [_ u / x ]_ B
13	11, 12	nfxp 4758	. . . 4 |- F/_ x ( { u } X. [_ u / x ]_ B )
14		sneq 3684	. . . . 5 |- ( x = u -> { x } = { u } )
15		csbeq1a 3137	. . . . 5 |- ( x = u -> B = [_ u / x ]_ B )
16	14, 15	xpeq12d 4756	. . . 4 |- ( x = u -> ( { x } X. B ) = ( { u } X. [_ u / x ]_ B ) )
17	10, 13, 16	cbviun 4012	. . 3 |- U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B )
18		mpteq1 4178	. . 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 2375	. . 3 |- F/_ u B
21		nfcv 2375	. . 3 |- F/_ u C
22		nfcv 2375	. . 3 |- F/_ v C
23		nfcsb1v 3161	. . 3 |- F/_ x [_ u / x ]_ [_ v / y ]_ C
24		nfcv 2375	. . . 4 |- F/_ y u
25		nfcsb1v 3161	. . . 4 |- F/_ y [_ v / y ]_ C
26	24, 25	nfcsb 3166	. . 3 |- F/_ y [_ u / x ]_ [_ v / y ]_ C
27		csbeq1a 3137	. . . 4 |- ( y = v -> C = [_ v / y ]_ C )
28		csbeq1a 3137	. . . 4 |- ( x = u -> [_ v / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
29	27, 28	sylan9eqr 2286	. . 3 |- ( ( x = u /\ y = v ) -> C = [_ u / x ]_ [_ v / y ]_ C )
30	20, 12, 21, 22, 23, 26, 15, 29	cbvmpox 6109	. 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 2263	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 1398   [_csb 3128   {csn 3673   <.cop 3676   U_ciun 3975   |-> cmpt 4155   X. cxp 4729   ` cfv 5333   e. cmpo 6030   1stc1st 6310   2ndc2nd 6311

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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fv 5341  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313

This theorem is referenced by:  mpompts  6372  mpofvex  6379