Theorem dmmpt2ssx2 30876

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 6750. (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 2616	. . . . 5 |- F/_ u A
2		nfcsb1v 3412	. . . . 5 |- F/_ y [_ u / y ]_ A
3		nfcv 2616	. . . . 5 |- F/_ u C
4		nfcv 2616	. . . . 5 |- F/_ v C
5		nfcv 2616	. . . . . 6 |- F/_ x u
6		nfcsb1v 3412	. . . . . 6 |- F/_ x [_ v / x ]_ C
7	5, 6	nfcsb 3414	. . . . 5 |- F/_ x [_ u / y ]_ [_ v / x ]_ C
8		nfcsb1v 3412	. . . . 5 |- F/_ y [_ u / y ]_ [_ v / x ]_ C
9		csbeq1a 3405	. . . . 5 |- ( y = u -> A = [_ u / y ]_ A )
10		csbeq1a 3405	. . . . . 6 |- ( x = v -> C = [_ v / x ]_ C )
11		csbeq1a 3405	. . . . . 6 |- ( y = u -> [_ v / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
12	10, 11	sylan9eqr 2517	. . . . 5 |- ( ( y = u /\ x = v ) -> C = [_ u / y ]_ [_ v / x ]_ C )
13	1, 2, 3, 4, 7, 8, 9, 12	cbvmpt2x2 30875	. . . 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 3081	. . . . . . . 8 |- v e. _V
16		vex 3081	. . . . . . . 8 |- u e. _V
17	15, 16	op2ndd 6699	. . . . . . 7 |- ( t = <. v , u >. -> ( 2nd ` t ) = u )
18	17	csbeq1d 3403	. . . . . 6 |- ( t = <. v , u >. -> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ ( 1st ` t ) / x ]_ C )
19	15, 16	op1std 6698	. . . . . . . 8 |- ( t = <. v , u >. -> ( 1st ` t ) = v )
20	19	csbeq1d 3403	. . . . . . 7 |- ( t = <. v , u >. -> [_ ( 1st ` t ) / x ]_ C = [_ v / x ]_ C )
21	20	csbeq2dv 3796	. . . . . 6 |- ( t = <. v , u >. -> [_ u / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
22	18, 21	eqtrd 2495	. . . . 5 |- ( t = <. v , u >. -> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
23	22	mpt2mptx2 30873	. . . 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 2493	. . 3 |- F = ( t e. U_ u e. B ( [_ u / y ]_ A X. { u } ) |-> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C )
25	24	dmmptss 5443	. 2 |- dom F C_ U_ u e. B ( [_ u / y ]_ A X. { u } )
26		nfcv 2616	. . 3 |- F/_ u ( A X. { y } )
27		nfcv 2616	. . . 4 |- F/_ y { u }
28	2, 27	nfxp 4975	. . 3 |- F/_ y ( [_ u / y ]_ A X. { u } )
29		sneq 3996	. . . 4 |- ( y = u -> { y } = { u } )
30	9, 29	xpeq12d 4974	. . 3 |- ( y = u -> ( A X. { y } ) = ( [_ u / y ]_ A X. { u } ) )
31	26, 28, 30	cbviun 4316	. 2 |- U_ y e. B ( A X. { y } ) = U_ u e. B ( [_ u / y ]_ A X. { u } )
32	25, 31	sseqtr4i 3498	1 |- dom F C_ U_ y e. B ( A X. { y } )

Syntax hints:   = wceq 1370   [_csb 3396   C_ wss 3437   {csn 3986   <.cop 3992   U_ciun 4280   |-> cmpt 4459   X. cxp 4947   dom cdm 4949   ` cfv 5527   |-> cmpt2 6203   1stc1st 6686   2ndc2nd 6687

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 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

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 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fv 5535  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689

This theorem is referenced by:  mpt2exxg2  30877