Theorem dmmpossx 6167

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 2308	. . . . 5 |- F/_ u B
2		nfcsb1v 3078	. . . . 5 |- F/_ x [_ u / x ]_ B
3		nfcv 2308	. . . . 5 |- F/_ u C
4		nfcv 2308	. . . . 5 |- F/_ v C
5		nfcsb1v 3078	. . . . 5 |- F/_ x [_ u / x ]_ [_ v / y ]_ C
6		nfcv 2308	. . . . . 6 |- F/_ y u
7		nfcsb1v 3078	. . . . . 6 |- F/_ y [_ v / y ]_ C
8	6, 7	nfcsb 3082	. . . . 5 |- F/_ y [_ u / x ]_ [_ v / y ]_ C
9		csbeq1a 3054	. . . . 5 |- ( x = u -> B = [_ u / x ]_ B )
10		csbeq1a 3054	. . . . . 6 |- ( y = v -> C = [_ v / y ]_ C )
11		csbeq1a 3054	. . . . . 6 |- ( x = u -> [_ v / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
12	10, 11	sylan9eqr 2221	. . . . 5 |- ( ( x = u /\ y = v ) -> C = [_ u / x ]_ [_ v / y ]_ C )
13	1, 2, 3, 4, 5, 8, 9, 12	cbvmpox 5920	. . . 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 2729	. . . . . . . 8 |- u e. _V
16		vex 2729	. . . . . . . 8 |- v e. _V
17	15, 16	op1std 6116	. . . . . . 7 |- ( t = <. u , v >. -> ( 1st ` t ) = u )
18	17	csbeq1d 3052	. . . . . 6 |- ( t = <. u , v >. -> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ ( 2nd ` t ) / y ]_ C )
19	15, 16	op2ndd 6117	. . . . . . . 8 |- ( t = <. u , v >. -> ( 2nd ` t ) = v )
20	19	csbeq1d 3052	. . . . . . 7 |- ( t = <. u , v >. -> [_ ( 2nd ` t ) / y ]_ C = [_ v / y ]_ C )
21	20	csbeq2dv 3071	. . . . . 6 |- ( t = <. u , v >. -> [_ u / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
22	18, 21	eqtrd 2198	. . . . 5 |- ( t = <. u , v >. -> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
23	22	mpomptx 5933	. . . 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 2196	. . 3 |- F = ( t e. U_ u e. A ( { u } X. [_ u / x ]_ B ) |-> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C )
25	24	dmmptss 5100	. 2 |- dom F C_ U_ u e. A ( { u } X. [_ u / x ]_ B )
26		nfcv 2308	. . 3 |- F/_ u ( { x } X. B )
27		nfcv 2308	. . . 4 |- F/_ x { u }
28	27, 2	nfxp 4631	. . 3 |- F/_ x ( { u } X. [_ u / x ]_ B )
29		sneq 3587	. . . 4 |- ( x = u -> { x } = { u } )
30	29, 9	xpeq12d 4629	. . 3 |- ( x = u -> ( { x } X. B ) = ( { u } X. [_ u / x ]_ B ) )
31	26, 28, 30	cbviun 3903	. 2 |- U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B )
32	25, 31	sseqtrri 3177	1 |- dom F C_ U_ x e. A ( { x } X. B )

Syntax hints:   = wceq 1343   [_csb 3045   C_ wss 3116   {csn 3576   <.cop 3579   U_ciun 3866   |-> cmpt 4043   X. cxp 4602   dom cdm 4604   ` cfv 5188   e. cmpo 5844   1stc1st 6106   2ndc2nd 6107

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fv 5196  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109

This theorem is referenced by:  mpoexxg  6178  mpoxopn0yelv  6207