Theorem fvmptssdm 5553

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 5469	. . . . . 6 |- ( y = D -> ( F ` y ) = ( F ` D ) )
2	1	sseq1d 3157	. . . . 5 |- ( y = D -> ( ( F ` y ) C_ C <-> ( F ` D ) C_ C ) )
3	2	imbi2d 229	. . . 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 2636	. . . . . . 7 |- F/_ x { x e. A | B e. _V }
5	4	nfcri 2293	. . . . . 6 |- F/ x y e. { x e. A | B e. _V }
6		nfra1 2488	. . . . . . 7 |- F/ x A. x e. A B C_ C
7		fvmpt2.1	. . . . . . . . . 10 |- F = ( x e. A |-> B )
8		nfmpt1 4058	. . . . . . . . . 10 |- F/_ x ( x e. A |-> B )
9	7, 8	nfcxfr 2296	. . . . . . . . 9 |- F/_ x F
10		nfcv 2299	. . . . . . . . 9 |- F/_ x y
11	9, 10	nffv 5479	. . . . . . . 8 |- F/_ x ( F ` y )
12		nfcv 2299	. . . . . . . 8 |- F/_ x C
13	11, 12	nfss 3121	. . . . . . 7 |- F/ x ( F ` y ) C_ C
14	6, 13	nfim 1552	. . . . . 6 |- F/ x ( A. x e. A B C_ C -> ( F ` y ) C_ C )
15	5, 14	nfim 1552	. . . . 5 |- F/ x ( y e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` y ) C_ C ) )
16		eleq1 2220	. . . . . 6 |- ( x = y -> ( x e. { x e. A | B e. _V } <-> y e. { x e. A | B e. _V } ) )
17		fveq2 5469	. . . . . . . 8 |- ( x = y -> ( F ` x ) = ( F ` y ) )
18	17	sseq1d 3157	. . . . . . 7 |- ( x = y -> ( ( F ` x ) C_ C <-> ( F ` y ) C_ C ) )
19	18	imbi2d 229	. . . . . 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 233	. . . . 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 5082	. . . . . . 7 |- dom F = { x e. A | B e. _V }
22	21	eleq2i 2224	. . . . . 6 |- ( x e. dom F <-> x e. { x e. A | B e. _V } )
23	21	rabeq2i 2709	. . . . . . . . . 10 |- ( x e. dom F <-> ( x e. A /\ B e. _V ) )
24	7	fvmpt2 5552	. . . . . . . . . . 11 |- ( ( x e. A /\ B e. _V ) -> ( F ` x ) = B )
25		eqimss 3182	. . . . . . . . . . 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 120	. . . . . . . . 9 |- ( x e. dom F -> ( F ` x ) C_ B )
28	27	adantr 274	. . . . . . . 8 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> ( F ` x ) C_ B )
29	7	dmmptss 5083	. . . . . . . . . 10 |- dom F C_ A
30	29	sseli 3124	. . . . . . . . 9 |- ( x e. dom F -> x e. A )
31		rsp 2504	. . . . . . . . 9 |- ( A. x e. A B C_ C -> ( x e. A -> B C_ C ) )
32	30, 31	mpan9 279	. . . . . . . 8 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> B C_ C )
33	28, 32	sstrd 3138	. . . . . . 7 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> ( F ` x ) C_ C )
34	33	ex 114	. . . . . 6 |- ( x e. dom F -> ( A. x e. A B C_ C -> ( F ` x ) C_ C ) )
35	22, 34	sylbir 134	. . . . 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 1737	. . . 4 |- ( y e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` y ) C_ C ) )
37	3, 36	vtoclga 2778	. . 3 |- ( D e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` D ) C_ C ) )
38	37, 21	eleq2s 2252	. 2 |- ( D e. dom F -> ( A. x e. A B C_ C -> ( F ` D ) C_ C ) )
39	38	imp 123	1 |- ( ( D e. dom F /\ A. x e. A B C_ C ) -> ( F ` D ) C_ C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   A.wral 2435   {crab 2439   _Vcvv 2712   C_ wss 3102   |-> cmpt 4026   dom cdm 4587   ` cfv 5171

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fv 5179

This theorem is referenced by: (None)