Theorem dmmpossx 6178

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

Hypothesis

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

Assertion

Ref	Expression
dmmpossx	|- 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 dmmpossx

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

Step	Hyp	Ref	Expression
1		nfcv 2312	. . . . 5 |- F/_ u B
2		nfcsb1v 3082	. . . . 5 |- F/_ x [_ u / x ]_ B
3		nfcv 2312	. . . . 5 |- F/_ u C
4		nfcv 2312	. . . . 5 |- F/_ v C
5		nfcsb1v 3082	. . . . 5 |- F/_ x [_ u / x ]_ [_ v / y ]_ C
6		nfcv 2312	. . . . . 6 |- F/_ y u
7		nfcsb1v 3082	. . . . . 6 |- F/_ y [_ v / y ]_ C
8	6, 7	nfcsb 3086	. . . . 5 |- F/_ y [_ u / x ]_ [_ v / y ]_ C
9		csbeq1a 3058	. . . . 5 |- ( x = u -> B = [_ u / x ]_ B )
10		csbeq1a 3058	. . . . . 6 |- ( y = v -> C = [_ v / y ]_ C )
11		csbeq1a 3058	. . . . . 6 |- ( x = u -> [_ v / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
12	10, 11	sylan9eqr 2225	. . . . 5 |- ( ( x = u /\ y = v ) -> C = [_ u / x ]_ [_ v / y ]_ C )
13	1, 2, 3, 4, 5, 8, 9, 12	cbvmpox 5931	. . . 4 |- ( x e. A , y e. B |-> C ) = ( u e. A , v e. [_ u / x ]_ B |-> [_ u / x ]_ [_ v / y ]_ C )
14		fmpox.1	. . . 4 |- F = ( x e. A , y e. B |-> C )
15		vex 2733	. . . . . . . 8 |- u e. _V
16		vex 2733	. . . . . . . 8 |- v e. _V
17	15, 16	op1std 6127	. . . . . . 7 |- ( t = <. u , v >. -> ( 1st ` t ) = u )
18	17	csbeq1d 3056	. . . . . 6 |- ( t = <. u , v >. -> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ ( 2nd ` t ) / y ]_ C )
19	15, 16	op2ndd 6128	. . . . . . . 8 |- ( t = <. u , v >. -> ( 2nd ` t ) = v )
20	19	csbeq1d 3056	. . . . . . 7 |- ( t = <. u , v >. -> [_ ( 2nd ` t ) / y ]_ C = [_ v / y ]_ C )
21	20	csbeq2dv 3075	. . . . . 6 |- ( t = <. u , v >. -> [_ u / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
22	18, 21	eqtrd 2203	. . . . 5 |- ( t = <. u , v >. -> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
23	22	mpomptx 5944	. . . 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 2201	. . 3 |- F = ( t e. U_ u e. A ( { u } X. [_ u / x ]_ B ) |-> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C )
25	24	dmmptss 5107	. 2 |- dom F C_ U_ u e. A ( { u } X. [_ u / x ]_ B )
26		nfcv 2312	. . 3 |- F/_ u ( { x } X. B )
27		nfcv 2312	. . . 4 |- F/_ x { u }
28	27, 2	nfxp 4638	. . 3 |- F/_ x ( { u } X. [_ u / x ]_ B )
29		sneq 3594	. . . 4 |- ( x = u -> { x } = { u } )
30	29, 9	xpeq12d 4636	. . 3 |- ( x = u -> ( { x } X. B ) = ( { u } X. [_ u / x ]_ B ) )
31	26, 28, 30	cbviun 3910	. 2 |- U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B )
32	25, 31	sseqtrri 3182	1 |- dom F C_ U_ x e. A ( { x } X. B )

Syntax hints:   = wceq 1348   [_csb 3049   C_ wss 3121   {csn 3583   <.cop 3586   U_ciun 3873   |-> cmpt 4050   X. cxp 4609   dom cdm 4611   ` cfv 5198   e. cmpo 5855   1stc1st 6117   2ndc2nd 6118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fv 5206  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120

This theorem is referenced by:  mpoexxg  6189  mpoxopn0yelv  6218