Theorem dmmpossx 6097

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 2281	. . . . 5 |- F/_ u B
2		nfcsb1v 3035	. . . . 5 |- F/_ x [_ u / x ]_ B
3		nfcv 2281	. . . . 5 |- F/_ u C
4		nfcv 2281	. . . . 5 |- F/_ v C
5		nfcsb1v 3035	. . . . 5 |- F/_ x [_ u / x ]_ [_ v / y ]_ C
6		nfcv 2281	. . . . . 6 |- F/_ y u
7		nfcsb1v 3035	. . . . . 6 |- F/_ y [_ v / y ]_ C
8	6, 7	nfcsb 3037	. . . . 5 |- F/_ y [_ u / x ]_ [_ v / y ]_ C
9		csbeq1a 3012	. . . . 5 |- ( x = u -> B = [_ u / x ]_ B )
10		csbeq1a 3012	. . . . . 6 |- ( y = v -> C = [_ v / y ]_ C )
11		csbeq1a 3012	. . . . . 6 |- ( x = u -> [_ v / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
12	10, 11	sylan9eqr 2194	. . . . 5 |- ( ( x = u /\ y = v ) -> C = [_ u / x ]_ [_ v / y ]_ C )
13	1, 2, 3, 4, 5, 8, 9, 12	cbvmpox 5849	. . . 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 2689	. . . . . . . 8 |- u e. _V
16		vex 2689	. . . . . . . 8 |- v e. _V
17	15, 16	op1std 6046	. . . . . . 7 |- ( t = <. u , v >. -> ( 1st ` t ) = u )
18	17	csbeq1d 3010	. . . . . 6 |- ( t = <. u , v >. -> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ ( 2nd ` t ) / y ]_ C )
19	15, 16	op2ndd 6047	. . . . . . . 8 |- ( t = <. u , v >. -> ( 2nd ` t ) = v )
20	19	csbeq1d 3010	. . . . . . 7 |- ( t = <. u , v >. -> [_ ( 2nd ` t ) / y ]_ C = [_ v / y ]_ C )
21	20	csbeq2dv 3028	. . . . . 6 |- ( t = <. u , v >. -> [_ u / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
22	18, 21	eqtrd 2172	. . . . 5 |- ( t = <. u , v >. -> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
23	22	mpomptx 5862	. . . 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 2170	. . 3 |- F = ( t e. U_ u e. A ( { u } X. [_ u / x ]_ B ) |-> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C )
25	24	dmmptss 5035	. 2 |- dom F C_ U_ u e. A ( { u } X. [_ u / x ]_ B )
26		nfcv 2281	. . 3 |- F/_ u ( { x } X. B )
27		nfcv 2281	. . . 4 |- F/_ x { u }
28	27, 2	nfxp 4566	. . 3 |- F/_ x ( { u } X. [_ u / x ]_ B )
29		sneq 3538	. . . 4 |- ( x = u -> { x } = { u } )
30	29, 9	xpeq12d 4564	. . 3 |- ( x = u -> ( { x } X. B ) = ( { u } X. [_ u / x ]_ B ) )
31	26, 28, 30	cbviun 3850	. 2 |- U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B )
32	25, 31	sseqtrri 3132	1 |- dom F C_ U_ x e. A ( { x } X. B )

Syntax hints:   = wceq 1331   [_csb 3003   C_ wss 3071   {csn 3527   <.cop 3530   U_ciun 3813   |-> cmpt 3989   X. cxp 4537   dom cdm 4539   ` cfv 5123   e. cmpo 5776   1stc1st 6036   2ndc2nd 6037

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fv 5131  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039

This theorem is referenced by:  mpoexxg  6108  mpoxopn0yelv  6136