Theorem dmmpt2ssx2 33070

Description: The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpt2ssx 6864. (Contributed by AV, 30-Mar-2019.)

Hypothesis

Ref	Expression
dmmpt2ssx2.1	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
dmmpt2ssx2	|- dom F C_ U_ y e. B ( A X. { y } )

Distinct variable groups:   x, A   x, y, B

Allowed substitution hints:   A( y)   C( x, y)   F( x, y)

Proof of Theorem dmmpt2ssx2

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

Step	Hyp	Ref	Expression
1		nfcv 2619	. . . . 5 |- F/_ u A
2		nfcsb1v 3446	. . . . 5 |- F/_ y [_ u / y ]_ A
3		nfcv 2619	. . . . 5 |- F/_ u C
4		nfcv 2619	. . . . 5 |- F/_ v C
5		nfcv 2619	. . . . . 6 |- F/_ x u
6		nfcsb1v 3446	. . . . . 6 |- F/_ x [_ v / x ]_ C
7	5, 6	nfcsb 3448	. . . . 5 |- F/_ x [_ u / y ]_ [_ v / x ]_ C
8		nfcsb1v 3446	. . . . 5 |- F/_ y [_ u / y ]_ [_ v / x ]_ C
9		csbeq1a 3439	. . . . 5 |- ( y = u -> A = [_ u / y ]_ A )
10		csbeq1a 3439	. . . . . 6 |- ( x = v -> C = [_ v / x ]_ C )
11		csbeq1a 3439	. . . . . 6 |- ( y = u -> [_ v / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
12	10, 11	sylan9eqr 2520	. . . . 5 |- ( ( y = u /\ x = v ) -> C = [_ u / y ]_ [_ v / x ]_ C )
13	1, 2, 3, 4, 7, 8, 9, 12	cbvmpt2x2 33069	. . . 4 |- ( x e. A , y e. B |-> C ) = ( v e. [_ u / y ]_ A , u e. B |-> [_ u / y ]_ [_ v / x ]_ C )
14		dmmpt2ssx2.1	. . . 4 |- F = ( x e. A , y e. B |-> C )
15		vex 3112	. . . . . . . 8 |- v e. _V
16		vex 3112	. . . . . . . 8 |- u e. _V
17	15, 16	op2ndd 6810	. . . . . . 7 |- ( t = <. v , u >. -> ( 2nd ` t ) = u )
18	17	csbeq1d 3437	. . . . . 6 |- ( t = <. v , u >. -> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ ( 1st ` t ) / x ]_ C )
19	15, 16	op1std 6809	. . . . . . . 8 |- ( t = <. v , u >. -> ( 1st ` t ) = v )
20	19	csbeq1d 3437	. . . . . . 7 |- ( t = <. v , u >. -> [_ ( 1st ` t ) / x ]_ C = [_ v / x ]_ C )
21	20	csbeq2dv 3843	. . . . . 6 |- ( t = <. v , u >. -> [_ u / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
22	18, 21	eqtrd 2498	. . . . 5 |- ( t = <. v , u >. -> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
23	22	mpt2mptx2 33068	. . . 4 |- ( t e. U_ u e. B ( [_ u / y ]_ A X. { u } ) |-> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C ) = ( v e. [_ u / y ]_ A , u e. B |-> [_ u / y ]_ [_ v / x ]_ C )
24	13, 14, 23	3eqtr4i 2496	. . 3 |- F = ( t e. U_ u e. B ( [_ u / y ]_ A X. { u } ) |-> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C )
25	24	dmmptss 5509	. 2 |- dom F C_ U_ u e. B ( [_ u / y ]_ A X. { u } )
26		nfcv 2619	. . 3 |- F/_ u ( A X. { y } )
27		nfcv 2619	. . . 4 |- F/_ y { u }
28	2, 27	nfxp 5035	. . 3 |- F/_ y ( [_ u / y ]_ A X. { u } )
29		sneq 4042	. . . 4 |- ( y = u -> { y } = { u } )
30	9, 29	xpeq12d 5033	. . 3 |- ( y = u -> ( A X. { y } ) = ( [_ u / y ]_ A X. { u } ) )
31	26, 28, 30	cbviun 4369	. 2 |- U_ y e. B ( A X. { y } ) = U_ u e. B ( [_ u / y ]_ A X. { u } )
32	25, 31	sseqtr4i 3532	1 |- dom F C_ U_ y e. B ( A X. { y } )

Syntax hints:   = wceq 1395   [_csb 3430   C_ wss 3471   {csn 4032   <.cop 4038   U_ciun 4332   |-> cmpt 4515   X. cxp 5006   dom cdm 5008   ` cfv 5594   |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800

This theorem is referenced by:  mpt2exxg2  33071