Theorem mpt2mptsx 5983

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 2625	. . . . . 6 |- u e. _V
2		vex 2625	. . . . . 6 |- v e. _V
3	1, 2	op1std 5935	. . . . 5 |- ( z = <. u , v >. -> ( 1st ` z ) = u )
4	3	csbeq1d 2942	. . . 4 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C )
5	1, 2	op2ndd 5936	. . . . . 6 |- ( z = <. u , v >. -> ( 2nd ` z ) = v )
6	5	csbeq1d 2942	. . . . 5 |- ( z = <. u , v >. -> [_ ( 2nd ` z ) / y ]_ C = [_ v / y ]_ C )
7	6	csbeq2dv 2959	. . . 4 |- ( z = <. u , v >. -> [_ u / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
8	4, 7	eqtrd 2121	. . 3 |- ( z = <. u , v >. -> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
9	8	mpt2mptx 5755	. 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 2229	. . . 4 |- F/_ u ( { x } X. B )
11		nfcv 2229	. . . . 5 |- F/_ x { u }
12		nfcsb1v 2966	. . . . 5 |- F/_ x [_ u / x ]_ B
13	11, 12	nfxp 4480	. . . 4 |- F/_ x ( { u } X. [_ u / x ]_ B )
14		sneq 3463	. . . . 5 |- ( x = u -> { x } = { u } )
15		csbeq1a 2944	. . . . 5 |- ( x = u -> B = [_ u / x ]_ B )
16	14, 15	xpeq12d 4479	. . . 4 |- ( x = u -> ( { x } X. B ) = ( { u } X. [_ u / x ]_ B ) )
17	10, 13, 16	cbviun 3775	. . 3 |- U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B )
18		mpteq1 3930	. . 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 7	. 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 2229	. . 3 |- F/_ u B
21		nfcv 2229	. . 3 |- F/_ u C
22		nfcv 2229	. . 3 |- F/_ v C
23		nfcsb1v 2966	. . 3 |- F/_ x [_ u / x ]_ [_ v / y ]_ C
24		nfcv 2229	. . . 4 |- F/_ y u
25		nfcsb1v 2966	. . . 4 |- F/_ y [_ v / y ]_ C
26	24, 25	nfcsb 2968	. . 3 |- F/_ y [_ u / x ]_ [_ v / y ]_ C
27		csbeq1a 2944	. . . 4 |- ( y = v -> C = [_ v / y ]_ C )
28		csbeq1a 2944	. . . 4 |- ( x = u -> [_ v / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
29	27, 28	sylan9eqr 2143	. . 3 |- ( ( x = u /\ y = v ) -> C = [_ u / x ]_ [_ v / y ]_ C )
30	20, 12, 21, 22, 23, 26, 15, 29	cbvmpt2x 5742	. 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 2120	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 1290   [_csb 2936   {csn 3452   <.cop 3455   U_ciun 3738   |-> cmpt 3907   X. cxp 4452   ` cfv 5030   |-> cmpt2 5670   1stc1st 5925   2ndc2nd 5926

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-csb 2937  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-iota 4995  df-fun 5032  df-fv 5038  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928

This theorem is referenced by:  mpt2mpts  5984  mpt2fvex  5989