Theorem mpt2mptsx 6857

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

Assertion

Ref	Expression
mpt2mptsx	|- ( 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 mpt2mptsx

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

Step	Hyp	Ref	Expression
1		vex 3121	. . . . . 6 |- u e. _V
2		vex 3121	. . . . . 6 |- v e. _V
3	1, 2	op1std 6804	. . . . 5 |- ( z = <. u , v >. -> ( 1st ` z ) = u )
4	3	csbeq1d 3447	. . . 4 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C )
5	1, 2	op2ndd 6805	. . . . . 6 |- ( z = <. u , v >. -> ( 2nd ` z ) = v )
6	5	csbeq1d 3447	. . . . 5 |- ( z = <. u , v >. -> [_ ( 2nd ` z ) / y ]_ C = [_ v / y ]_ C )
7	6	csbeq2dv 3840	. . . 4 |- ( z = <. u , v >. -> [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
8	4, 7	eqtrd 2508	. . 3 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
9	8	mpt2mptx 6387	. 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 2629	. . . 4 |- F/_ u ( { x } X. B )
11		nfcv 2629	. . . . 5 |- F/_ x { u }
12		nfcsb1v 3456	. . . . 5 |- F/_ x [_ u / x ]_ B
13	11, 12	nfxp 5031	. . . 4 |- F/_ x ( { u } X. [_ u / x ]_ B )
14		sneq 4042	. . . . 5 |- ( x = u -> { x } = { u } )
15		csbeq1a 3449	. . . . 5 |- ( x = u -> B = [_ u / x ]_ B )
16	14, 15	xpeq12d 5029	. . . 4 |- ( x = u -> ( { x } X. B ) = ( { u } X. [_ u / x ]_ B ) )
17	10, 13, 16	cbviun 4367	. . 3 |- U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B )
18		mpteq1 4532	. . 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 2629	. . 3 |- F/_ u B
21		nfcv 2629	. . 3 |- F/_ u C
22		nfcv 2629	. . 3 |- F/_ v C
23		nfcsb1v 3456	. . 3 |- F/_ x [_ u / x ]_ [_ v / y ]_ C
24		nfcv 2629	. . . 4 |- F/_ y u
25		nfcsb1v 3456	. . . 4 |- F/_ y [_ v / y ]_ C
26	24, 25	nfcsb 3458	. . 3 |- F/_ y [_ u / x ]_ [_ v / y ]_ C
27		csbeq1a 3449	. . . 4 |- ( y = v -> C = [_ v / y ]_ C )
28		csbeq1a 3449	. . . 4 |- ( x = u -> [_ v / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
29	27, 28	sylan9eqr 2530	. . 3 |- ( ( x = u /\ y = v ) -> C = [_ u / x ]_ [_ v / y ]_ C )
30	20, 12, 21, 22, 23, 26, 15, 29	cbvmpt2x 6369	. 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 2507	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 1379   [_csb 3440   {csn 4032   <.cop 4038   U_ciun 4330   |-> cmpt 4510   X. cxp 5002   ` cfv 5593   |-> cmpt2 6296   1stc1st 6792   2ndc2nd 6793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-oprab 6298  df-mpt2 6299  df-1st 6794  df-2nd 6795

This theorem is referenced by:  mpt2mpts  6858  ovmptss  6874