Theorem fvmptssdm 5276

Description: If all the values of the mapping are subsets of a class C, then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.)

Hypothesis

Ref	Expression
fvmpt2.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
fvmptssdm	|- ( ( D e. dom F /\ A. x e. A B C_ C ) -> ( F ` D ) C_ C )

Distinct variable groups:   x, A   x, C

Allowed substitution hints:   B( x)   D( x)   F( x)

Proof of Theorem fvmptssdm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5198	. . . . . 6 |- ( y = D -> ( F ` y ) = ( F ` D ) )
2	1	sseq1d 3026	. . . . 5 |- ( y = D -> ( ( F ` y ) C_ C <-> ( F ` D ) C_ C ) )
3	2	imbi2d 228	. . . 4 |- ( y = D -> ( ( A. x e. A B C_ C -> ( F ` y ) C_ C ) <-> ( A. x e. A B C_ C -> ( F ` D ) C_ C ) ) )
4		nfrab1 2533	. . . . . . 7 |- F/_ x { x e. A | B e. _V }
5	4	nfcri 2213	. . . . . 6 |- F/ x y e. { x e. A | B e. _V }
6		nfra1 2397	. . . . . . 7 |- F/ x A. x e. A B C_ C
7		fvmpt2.1	. . . . . . . . . 10 |- F = ( x e. A |-> B )
8		nfmpt1 3871	. . . . . . . . . 10 |- F/_ x ( x e. A |-> B )
9	7, 8	nfcxfr 2216	. . . . . . . . 9 |- F/_ x F
10		nfcv 2219	. . . . . . . . 9 |- F/_ x y
11	9, 10	nffv 5205	. . . . . . . 8 |- F/_ x ( F ` y )
12		nfcv 2219	. . . . . . . 8 |- F/_ x C
13	11, 12	nfss 2992	. . . . . . 7 |- F/ x ( F ` y ) C_ C
14	6, 13	nfim 1504	. . . . . 6 |- F/ x ( A. x e. A B C_ C -> ( F ` y ) C_ C )
15	5, 14	nfim 1504	. . . . 5 |- F/ x ( y e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` y ) C_ C ) )
16		eleq1 2141	. . . . . 6 |- ( x = y -> ( x e. { x e. A | B e. _V } <-> y e. { x e. A | B e. _V } ) )
17		fveq2 5198	. . . . . . . 8 |- ( x = y -> ( F ` x ) = ( F ` y ) )
18	17	sseq1d 3026	. . . . . . 7 |- ( x = y -> ( ( F ` x ) C_ C <-> ( F ` y ) C_ C ) )
19	18	imbi2d 228	. . . . . 6 |- ( x = y -> ( ( A. x e. A B C_ C -> ( F ` x ) C_ C ) <-> ( A. x e. A B C_ C -> ( F ` y ) C_ C ) ) )
20	16, 19	imbi12d 232	. . . . 5 |- ( x = y -> ( ( x e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` x ) C_ C ) ) <-> ( y e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` y ) C_ C ) ) ) )
21	7	dmmpt 4836	. . . . . . 7 |- dom F = { x e. A | B e. _V }
22	21	eleq2i 2145	. . . . . 6 |- ( x e. dom F <-> x e. { x e. A | B e. _V } )
23	21	rabeq2i 2598	. . . . . . . . . 10 |- ( x e. dom F <-> ( x e. A /\ B e. _V ) )
24	7	fvmpt2 5275	. . . . . . . . . . 11 |- ( ( x e. A /\ B e. _V ) -> ( F ` x ) = B )
25		eqimss 3051	. . . . . . . . . . 11 |- ( ( F ` x ) = B -> ( F ` x ) C_ B )
26	24, 25	syl 14	. . . . . . . . . 10 |- ( ( x e. A /\ B e. _V ) -> ( F ` x ) C_ B )
27	23, 26	sylbi 119	. . . . . . . . 9 |- ( x e. dom F -> ( F ` x ) C_ B )
28	27	adantr 270	. . . . . . . 8 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> ( F ` x ) C_ B )
29	7	dmmptss 4837	. . . . . . . . . 10 |- dom F C_ A
30	29	sseli 2995	. . . . . . . . 9 |- ( x e. dom F -> x e. A )
31		rsp 2411	. . . . . . . . 9 |- ( A. x e. A B C_ C -> ( x e. A -> B C_ C ) )
32	30, 31	mpan9 275	. . . . . . . 8 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> B C_ C )
33	28, 32	sstrd 3009	. . . . . . 7 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> ( F ` x ) C_ C )
34	33	ex 113	. . . . . 6 |- ( x e. dom F -> ( A. x e. A B C_ C -> ( F ` x ) C_ C ) )
35	22, 34	sylbir 133	. . . . 5 |- ( x e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` x ) C_ C ) )
36	15, 20, 35	chvar 1680	. . . 4 |- ( y e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` y ) C_ C ) )
37	3, 36	vtoclga 2664	. . 3 |- ( D e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` D ) C_ C ) )
38	37, 21	eleq2s 2173	. 2 |- ( D e. dom F -> ( A. x e. A B C_ C -> ( F ` D ) C_ C ) )
39	38	imp 122	1 |- ( ( D e. dom F /\ A. x e. A B C_ C ) -> ( F ` D ) C_ C )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   {crab 2352   _Vcvv 2601   C_ wss 2973   |-> cmpt 3839   dom cdm 4363   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fv 4930

This theorem is referenced by: (None)