Theorem dmmpt2ssx 5825

Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)

Hypothesis

Ref	Expression
fmpt2x.1	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
dmmpt2ssx	|- dom F C_ U_ x e. A ( { x } X. B )

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

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

Proof of Theorem dmmpt2ssx

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

Step	Hyp	Ref	Expression
1		nfcv 2178	. . . . 5 |- F/_ u B
2		nfcsb1v 2882	. . . . 5 |- F/_ x [_ u / x ]_ B
3		nfcv 2178	. . . . 5 |- F/_ u C
4		nfcv 2178	. . . . 5 |- F/_ v C
5		nfcsb1v 2882	. . . . 5 |- F/_ x [_ u / x ]_ [_ v / y ]_ C
6		nfcv 2178	. . . . . 6 |- F/_ y u
7		nfcsb1v 2882	. . . . . 6 |- F/_ y [_ v / y ]_ C
8	6, 7	nfcsb 2884	. . . . 5 |- F/_ y [_ u / x ]_ [_ v / y ]_ C
9		csbeq1a 2860	. . . . 5 |- ( x = u -> B = [_ u / x ]_ B )
10		csbeq1a 2860	. . . . . 6 |- ( y = v -> C = [_ v / y ]_ C )
11		csbeq1a 2860	. . . . . 6 |- ( x = u -> [_ v / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
12	10, 11	sylan9eqr 2094	. . . . 5 |- ( ( x = u /\ y = v ) -> C = [_ u / x ]_ [_ v / y ]_ C )
13	1, 2, 3, 4, 5, 8, 9, 12	cbvmpt2x 5582	. . . 4 |- ( x e. A , y e. B |-> C ) = ( u e. A , v e. [_ u / x ]_ B |-> [_ u / x ]_ [_ v / y ]_ C )
14		fmpt2x.1	. . . 4 |- F = ( x e. A , y e. B |-> C )
15		vex 2560	. . . . . . . 8 |- u e. _V
16		vex 2560	. . . . . . . 8 |- v e. _V
17	15, 16	op1std 5775	. . . . . . 7 |- ( t = <. u , v >. -> ( 1st ` t ) = u )
18	17	csbeq1d 2858	. . . . . 6 |- ( t = <. u , v >. -> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ ( 2nd ` t ) / y ]_ C )
19	15, 16	op2ndd 5776	. . . . . . . 8 |- ( t = <. u , v >. -> ( 2nd ` t ) = v )
20	19	csbeq1d 2858	. . . . . . 7 |- ( t = <. u , v >. -> [_ ( 2nd ` t ) / y ]_ C = [_ v / y ]_ C )
21	20	csbeq2dv 2875	. . . . . 6 |- ( t = <. u , v >. -> [_ u / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
22	18, 21	eqtrd 2072	. . . . 5 |- ( t = <. u , v >. -> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
23	22	mpt2mptx 5595	. . . 4 |- ( t e. U_ u e. A ( { u } X. [_ u / x ]_ B ) |-> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C ) = ( u e. A , v e. [_ u / x ]_ B |-> [_ u / x ]_ [_ v / y ]_ C )
24	13, 14, 23	3eqtr4i 2070	. . 3 |- F = ( t e. U_ u e. A ( { u } X. [_ u / x ]_ B ) |-> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C )
25	24	dmmptss 4817	. 2 |- dom F C_ U_ u e. A ( { u } X. [_ u / x ]_ B )
26		nfcv 2178	. . 3 |- F/_ u ( { x } X. B )
27		nfcv 2178	. . . 4 |- F/_ x { u }
28	27, 2	nfxp 4371	. . 3 |- F/_ x ( { u } X. [_ u / x ]_ B )
29		sneq 3386	. . . 4 |- ( x = u -> { x } = { u } )
30	29, 9	xpeq12d 4370	. . 3 |- ( x = u -> ( { x } X. B ) = ( { u } X. [_ u / x ]_ B ) )
31	26, 28, 30	cbviun 3694	. 2 |- U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B )
32	25, 31	sseqtr4i 2978	1 |- dom F C_ U_ x e. A ( { x } X. B )

Syntax hints:   = wceq 1243   [_csb 2852   C_ wss 2917   {csn 3375   <.cop 3378   U_ciun 3657   |-> cmpt 3818   X. cxp 4343   dom cdm 4345   ` cfv 4902   |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fv 4910  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768

This theorem is referenced by:  mpt2exxg  5833  mpt2xopn0yelv  5854