Theorem mpt2mptsx 5765

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 2554	. . . . . 6 |- u e. _V
2		vex 2554	. . . . . 6 |- v e. _V
3	1, 2	op1std 5717	. . . . 5 |- (z = <.u , v >. -> ( 1st ` z) = u)
4	3	csbeq1d 2852	. . . 4 |- (z = <.u , v >. -> [_( 1st ` z) / x ]_ [_( 2nd ` z) / y ]_ C = [_u / x ]_ [_( 2nd ` z) / y ]_ C)
5	1, 2	op2ndd 5718	. . . . . 6 |- (z = <.u , v >. -> ( 2nd ` z) = v)
6	5	csbeq1d 2852	. . . . 5 |- (z = <.u , v >. -> [_( 2nd ` z) / y ]_ C = [_v / y ]_ C)
7	6	csbeq2dv 2869	. . . 4 |- (z = <.u , v >. -> [_u / x ]_ [_( 2nd ` z) / y ]_ C = [_u / x ]_ [_v / y ]_ C)
8	4, 7	eqtrd 2069	. . 3 |- (z = <.u , v >. -> [_( 1st ` z) / x ]_ [_( 2nd ` z) / y ]_ C = [_u / x ]_ [_v / y ]_ C)
9	8	mpt2mptx 5537	. 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 2175	. . . 4 |- F/_u( {x } X. B)
11		nfcv 2175	. . . . 5 |- F/_x {u }
12		nfcsb1v 2876	. . . . 5 |- F/_x [_u / x ]_B
13	11, 12	nfxp 4314	. . . 4 |- F/_x( {u } X. [_u / x ]_B)
14		sneq 3378	. . . . 5 |- (x = u -> {x } = {u })
15		csbeq1a 2854	. . . . 5 |- (x = u -> B = [_u / x ]_B)
16	14, 15	xpeq12d 4313	. . . 4 |- (x = u -> ( {x } X. B) = ( {u } X. [_u / x ]_B))
17	10, 13, 16	cbviun 3685	. . 3 |- U_x e. A ( {x } X. B) = U_u e. A ( {u } X. [_u / x ]_B)
18		mpteq1 3832	. . 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 2175	. . 3 |- F/_uB
21		nfcv 2175	. . 3 |- F/_u C
22		nfcv 2175	. . 3 |- F/_v C
23		nfcsb1v 2876	. . 3 |- F/_x [_u / x ]_ [_v / y ]_ C
24		nfcv 2175	. . . 4 |- F/_yu
25		nfcsb1v 2876	. . . 4 |- F/_y [_v / y ]_ C
26	24, 25	nfcsb 2878	. . 3 |- F/_y [_u / x ]_ [_v / y ]_ C
27		csbeq1a 2854	. . . 4 |- (y = v -> C = [_v / y ]_ C)
28		csbeq1a 2854	. . . 4 |- (x = u -> [_v / y ]_ C = [_u / x ]_ [_v / y ]_ C)
29	27, 28	sylan9eqr 2091	. . 3 |- ((x = u /\ y = v) -> C = [_u / x ]_ [_v / y ]_ C)
30	20, 12, 21, 22, 23, 26, 15, 29	cbvmpt2x 5524	. 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 2068	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 1242   [_csb 2846   {csn 3367   <.cop 3370   U_ciun 3648   |-> cmpt 3809   X. cxp 4286   ` cfv 4845   |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710

This theorem is referenced by:  mpt2mpts  5766  mpt2fvex  5771