Theorem dmmpt2ssx2 42115

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 7235. (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 2764	. . . . 5 |- F/_ u A
2		nfcsb1v 3549	. . . . 5 |- F/_ y [_ u / y ]_ A
3		nfcv 2764	. . . . 5 |- F/_ u C
4		nfcv 2764	. . . . 5 |- F/_ v C
5		nfcv 2764	. . . . . 6 |- F/_ x u
6		nfcsb1v 3549	. . . . . 6 |- F/_ x [_ v / x ]_ C
7	5, 6	nfcsb 3551	. . . . 5 |- F/_ x [_ u / y ]_ [_ v / x ]_ C
8		nfcsb1v 3549	. . . . 5 |- F/_ y [_ u / y ]_ [_ v / x ]_ C
9		csbeq1a 3542	. . . . 5 |- ( y = u -> A = [_ u / y ]_ A )
10		csbeq1a 3542	. . . . . 6 |- ( x = v -> C = [_ v / x ]_ C )
11		csbeq1a 3542	. . . . . 6 |- ( y = u -> [_ v / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
12	10, 11	sylan9eqr 2678	. . . . 5 |- ( ( y = u /\ x = v ) -> C = [_ u / y ]_ [_ v / x ]_ C )
13	1, 2, 3, 4, 7, 8, 9, 12	cbvmpt2x2 42114	. . . 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 3203	. . . . . . . 8 |- v e. _V
16		vex 3203	. . . . . . . 8 |- u e. _V
17	15, 16	op2ndd 7179	. . . . . . 7 |- ( t = <. v , u >. -> ( 2nd ` t ) = u )
18	17	csbeq1d 3540	. . . . . 6 |- ( t = <. v , u >. -> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ ( 1st ` t ) / x ]_ C )
19	15, 16	op1std 7178	. . . . . . . 8 |- ( t = <. v , u >. -> ( 1st ` t ) = v )
20	19	csbeq1d 3540	. . . . . . 7 |- ( t = <. v , u >. -> [_ ( 1st ` t ) / x ]_ C = [_ v / x ]_ C )
21	20	csbeq2dv 3992	. . . . . 6 |- ( t = <. v , u >. -> [_ u / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
22	18, 21	eqtrd 2656	. . . . 5 |- ( t = <. v , u >. -> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
23	22	mpt2mptx2 42113	. . . 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 2654	. . 3 |- F = ( t e. U_ u e. B ( [_ u / y ]_ A X. { u } ) |-> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C )
25	24	dmmptss 5631	. 2 |- dom F C_ U_ u e. B ( [_ u / y ]_ A X. { u } )
26		nfcv 2764	. . 3 |- F/_ u ( A X. { y } )
27		nfcv 2764	. . . 4 |- F/_ y { u }
28	2, 27	nfxp 5142	. . 3 |- F/_ y ( [_ u / y ]_ A X. { u } )
29		sneq 4187	. . . 4 |- ( y = u -> { y } = { u } )
30	9, 29	xpeq12d 5140	. . 3 |- ( y = u -> ( A X. { y } ) = ( [_ u / y ]_ A X. { u } ) )
31	26, 28, 30	cbviun 4557	. 2 |- U_ y e. B ( A X. { y } ) = U_ u e. B ( [_ u / y ]_ A X. { u } )
32	25, 31	sseqtr4i 3638	1 |- dom F C_ U_ y e. B ( A X. { y } )

Syntax hints:   = wceq 1483   [_csb 3533   C_ wss 3574   {csn 4177   <.cop 4183   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ` cfv 5888   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  mpt2exxg2  42116