Theorem dmmpt2ssx2 40626

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 6877. (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 2612	. . . . 5 |- F/_ u A
2		nfcsb1v 3365	. . . . 5 |- F/_ y [_ u / y ]_ A
3		nfcv 2612	. . . . 5 |- F/_ u C
4		nfcv 2612	. . . . 5 |- F/_ v C
5		nfcv 2612	. . . . . 6 |- F/_ x u
6		nfcsb1v 3365	. . . . . 6 |- F/_ x [_ v / x ]_ C
7	5, 6	nfcsb 3367	. . . . 5 |- F/_ x [_ u / y ]_ [_ v / x ]_ C
8		nfcsb1v 3365	. . . . 5 |- F/_ y [_ u / y ]_ [_ v / x ]_ C
9		csbeq1a 3358	. . . . 5 |- ( y = u -> A = [_ u / y ]_ A )
10		csbeq1a 3358	. . . . . 6 |- ( x = v -> C = [_ v / x ]_ C )
11		csbeq1a 3358	. . . . . 6 |- ( y = u -> [_ v / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
12	10, 11	sylan9eqr 2527	. . . . 5 |- ( ( y = u /\ x = v ) -> C = [_ u / y ]_ [_ v / x ]_ C )
13	1, 2, 3, 4, 7, 8, 9, 12	cbvmpt2x2 40625	. . . 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 3034	. . . . . . . 8 |- v e. _V
16		vex 3034	. . . . . . . 8 |- u e. _V
17	15, 16	op2ndd 6823	. . . . . . 7 |- ( t = <. v , u >. -> ( 2nd ` t ) = u )
18	17	csbeq1d 3356	. . . . . 6 |- ( t = <. v , u >. -> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ ( 1st ` t ) / x ]_ C )
19	15, 16	op1std 6822	. . . . . . . 8 |- ( t = <. v , u >. -> ( 1st ` t ) = v )
20	19	csbeq1d 3356	. . . . . . 7 |- ( t = <. v , u >. -> [_ ( 1st ` t ) / x ]_ C = [_ v / x ]_ C )
21	20	csbeq2dv 3785	. . . . . 6 |- ( t = <. v , u >. -> [_ u / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
22	18, 21	eqtrd 2505	. . . . 5 |- ( t = <. v , u >. -> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C = [_ u / y ]_ [_ v / x ]_ C )
23	22	mpt2mptx2 40624	. . . 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 2503	. . 3 |- F = ( t e. U_ u e. B ( [_ u / y ]_ A X. { u } ) |-> [_ ( 2nd ` t ) / y ]_ [_ ( 1st ` t ) / x ]_ C )
25	24	dmmptss 5338	. 2 |- dom F C_ U_ u e. B ( [_ u / y ]_ A X. { u } )
26		nfcv 2612	. . 3 |- F/_ u ( A X. { y } )
27		nfcv 2612	. . . 4 |- F/_ y { u }
28	2, 27	nfxp 4866	. . 3 |- F/_ y ( [_ u / y ]_ A X. { u } )
29		sneq 3969	. . . 4 |- ( y = u -> { y } = { u } )
30	9, 29	xpeq12d 4864	. . 3 |- ( y = u -> ( A X. { y } ) = ( [_ u / y ]_ A X. { u } ) )
31	26, 28, 30	cbviun 4306	. 2 |- U_ y e. B ( A X. { y } ) = U_ u e. B ( [_ u / y ]_ A X. { u } )
32	25, 31	sseqtr4i 3451	1 |- dom F C_ U_ y e. B ( A X. { y } )

Syntax hints:   = wceq 1452   [_csb 3349   C_ wss 3390   {csn 3959   <.cop 3965   U_ciun 4269   |-> cmpt 4454   X. cxp 4837   dom cdm 4839   ` cfv 5589   |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813

This theorem is referenced by:  mpt2exxg2  40627