Theorem dmmpossx 6363

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 2374	. . . . 5 |- F/_ u B
2		nfcsb1v 3160	. . . . 5 |- F/_ x [_ u / x ]_ B
3		nfcv 2374	. . . . 5 |- F/_ u C
4		nfcv 2374	. . . . 5 |- F/_ v C
5		nfcsb1v 3160	. . . . 5 |- F/_ x [_ u / x ]_ [_ v / y ]_ C
6		nfcv 2374	. . . . . 6 |- F/_ y u
7		nfcsb1v 3160	. . . . . 6 |- F/_ y [_ v / y ]_ C
8	6, 7	nfcsb 3165	. . . . 5 |- F/_ y [_ u / x ]_ [_ v / y ]_ C
9		csbeq1a 3136	. . . . 5 |- ( x = u -> B = [_ u / x ]_ B )
10		csbeq1a 3136	. . . . . 6 |- ( y = v -> C = [_ v / y ]_ C )
11		csbeq1a 3136	. . . . . 6 |- ( x = u -> [_ v / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
12	10, 11	sylan9eqr 2286	. . . . 5 |- ( ( x = u /\ y = v ) -> C = [_ u / x ]_ [_ v / y ]_ C )
13	1, 2, 3, 4, 5, 8, 9, 12	cbvmpox 6098	. . . 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 2805	. . . . . . . 8 |- u e. _V
16		vex 2805	. . . . . . . 8 |- v e. _V
17	15, 16	op1std 6310	. . . . . . 7 |- ( t = <. u , v >. -> ( 1st ` t ) = u )
18	17	csbeq1d 3134	. . . . . 6 |- ( t = <. u , v >. -> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ ( 2nd ` t ) / y ]_ C )
19	15, 16	op2ndd 6311	. . . . . . . 8 |- ( t = <. u , v >. -> ( 2nd ` t ) = v )
20	19	csbeq1d 3134	. . . . . . 7 |- ( t = <. u , v >. -> [_ ( 2nd ` t ) / y ]_ C = [_ v / y ]_ C )
21	20	csbeq2dv 3153	. . . . . 6 |- ( t = <. u , v >. -> [_ u / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
22	18, 21	eqtrd 2264	. . . . 5 |- ( t = <. u , v >. -> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C = [_ u / x ]_ [_ v / y ]_ C )
23	22	mpomptx 6111	. . . 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 2262	. . 3 |- F = ( t e. U_ u e. A ( { u } X. [_ u / x ]_ B ) |-> [_ ( 1st ` t ) / x ]_ [_ ( 2nd ` t ) / y ]_ C )
25	24	dmmptss 5233	. 2 |- dom F C_ U_ u e. A ( { u } X. [_ u / x ]_ B )
26		nfcv 2374	. . 3 |- F/_ u ( { x } X. B )
27		nfcv 2374	. . . 4 |- F/_ x { u }
28	27, 2	nfxp 4752	. . 3 |- F/_ x ( { u } X. [_ u / x ]_ B )
29		sneq 3680	. . . 4 |- ( x = u -> { x } = { u } )
30	29, 9	xpeq12d 4750	. . 3 |- ( x = u -> ( { x } X. B ) = ( { u } X. [_ u / x ]_ B ) )
31	26, 28, 30	cbviun 4007	. 2 |- U_ x e. A ( { x } X. B ) = U_ u e. A ( { u } X. [_ u / x ]_ B )
32	25, 31	sseqtrri 3262	1 |- dom F C_ U_ x e. A ( { x } X. B )

Syntax hints:   = wceq 1397   [_csb 3127   C_ wss 3200   {csn 3669   <.cop 3672   U_ciun 3970   |-> cmpt 4150   X. cxp 4723   dom cdm 4725   ` cfv 5326   e. cmpo 6019   1stc1st 6300   2ndc2nd 6301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fv 5334  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303

This theorem is referenced by:  mpoexxg  6374  mpoxopn0yelv  6404