Theorem mpt2mptsx 6750

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 3081	. . . . . 6 |- u e. _V
2		vex 3081	. . . . . 6 |- v e. _V
3	1, 2	op1std 6700	. . . . 5 |- ( z = <. u , v >. -> ( 1st ` z ) = u )
4	3	csbeq1d 3405	. . . 4 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C )
5	1, 2	op2ndd 6701	. . . . . 6 |- ( z = <. u , v >. -> ( 2nd ` z ) = v )
6	5	csbeq1d 3405	. . . . 5 |- ( z = <. u , v >. -> [_ ( 2nd ` z ) / y ]_ C = [_ v / y ]_ C )
7	6	csbeq2dv 3798	. . . 4 |- ( z = <. u , v >. -> [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
8	4, 7	eqtrd 2495	. . 3 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
9	8	mpt2mptx 6294	. 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 2616	. . . 4 |- F/_ u ( { x } X. B )
11		nfcv 2616	. . . . 5 |- F/_ x { u }
12		nfcsb1v 3414	. . . . 5 |- F/_ x [_ u / x ]_ B
13	11, 12	nfxp 4977	. . . 4 |- F/_ x ( { u } X. [_ u / x ]_ B )
14		sneq 3998	. . . . 5 |- ( x = u -> { x } = { u } )
15		csbeq1a 3407	. . . . 5 |- ( x = u -> B = [_ u / x ]_ B )
16	14, 15	xpeq12d 4976	. . . 4 |- ( x = u -> ( { x } X. B ) = ( { u } X. [_ u / x ]_ B ) )
17	10, 13, 16	cbviun 4318	. . 3 |- U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B )
18		mpteq1 4483	. . 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 2616	. . 3 |- F/_ u B
21		nfcv 2616	. . 3 |- F/_ u C
22		nfcv 2616	. . 3 |- F/_ v C
23		nfcsb1v 3414	. . 3 |- F/_ x [_ u / x ]_ [_ v / y ]_ C
24		nfcv 2616	. . . 4 |- F/_ y u
25		nfcsb1v 3414	. . . 4 |- F/_ y [_ v / y ]_ C
26	24, 25	nfcsb 3416	. . 3 |- F/_ y [_ u / x ]_ [_ v / y ]_ C
27		csbeq1a 3407	. . . 4 |- ( y = v -> C = [_ v / y ]_ C )
28		csbeq1a 3407	. . . 4 |- ( x = u -> [_ v / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
29	27, 28	sylan9eqr 2517	. . 3 |- ( ( x = u /\ y = v ) -> C = [_ u / x ]_ [_ v / y ]_ C )
30	20, 12, 21, 22, 23, 26, 15, 29	cbvmpt2x 6276	. 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 2494	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 1370   [_csb 3398   {csn 3988   <.cop 3994   U_ciun 4282   |-> cmpt 4461   X. cxp 4949   ` cfv 5529   |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fv 5537  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691

This theorem is referenced by:  mpt2mpts  6751  ovmptss  6767