Theorem fvmptssdm 5255

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 5178	. . . . . 6 |- ( y = D -> ( F ` y ) = ( F ` D ) )
2	1	sseq1d 2972	. . . . 5 |- ( y = D -> ( ( F ` y ) C_ C <-> ( F ` D ) C_ C ) )
3	2	imbi2d 219	. . . 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 2489	. . . . . . 7 |- F/_ x { x e. A | B e. _V }
5	4	nfcri 2172	. . . . . 6 |- F/ x y e. { x e. A | B e. _V }
6		nfra1 2355	. . . . . . 7 |- F/ x A. x e. A B C_ C
7		fvmpt2.1	. . . . . . . . . 10 |- F = ( x e. A |-> B )
8		nfmpt1 3850	. . . . . . . . . 10 |- F/_ x ( x e. A |-> B )
9	7, 8	nfcxfr 2175	. . . . . . . . 9 |- F/_ x F
10		nfcv 2178	. . . . . . . . 9 |- F/_ x y
11	9, 10	nffv 5185	. . . . . . . 8 |- F/_ x ( F ` y )
12		nfcv 2178	. . . . . . . 8 |- F/_ x C
13	11, 12	nfss 2938	. . . . . . 7 |- F/ x ( F ` y ) C_ C
14	6, 13	nfim 1464	. . . . . 6 |- F/ x ( A. x e. A B C_ C -> ( F ` y ) C_ C )
15	5, 14	nfim 1464	. . . . 5 |- F/ x ( y e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` y ) C_ C ) )
16		eleq1 2100	. . . . . 6 |- ( x = y -> ( x e. { x e. A | B e. _V } <-> y e. { x e. A | B e. _V } ) )
17		fveq2 5178	. . . . . . . 8 |- ( x = y -> ( F ` x ) = ( F ` y ) )
18	17	sseq1d 2972	. . . . . . 7 |- ( x = y -> ( ( F ` x ) C_ C <-> ( F ` y ) C_ C ) )
19	18	imbi2d 219	. . . . . 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 223	. . . . 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 4816	. . . . . . 7 |- dom F = { x e. A | B e. _V }
22	21	eleq2i 2104	. . . . . 6 |- ( x e. dom F <-> x e. { x e. A | B e. _V } )
23	21	rabeq2i 2554	. . . . . . . . . 10 |- ( x e. dom F <-> ( x e. A /\ B e. _V ) )
24	7	fvmpt2 5254	. . . . . . . . . . 11 |- ( ( x e. A /\ B e. _V ) -> ( F ` x ) = B )
25		eqimss 2997	. . . . . . . . . . 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 114	. . . . . . . . 9 |- ( x e. dom F -> ( F ` x ) C_ B )
28	27	adantr 261	. . . . . . . 8 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> ( F ` x ) C_ B )
29	7	dmmptss 4817	. . . . . . . . . 10 |- dom F C_ A
30	29	sseli 2941	. . . . . . . . 9 |- ( x e. dom F -> x e. A )
31		rsp 2369	. . . . . . . . 9 |- ( A. x e. A B C_ C -> ( x e. A -> B C_ C ) )
32	30, 31	mpan9 265	. . . . . . . 8 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> B C_ C )
33	28, 32	sstrd 2955	. . . . . . 7 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> ( F ` x ) C_ C )
34	33	ex 108	. . . . . 6 |- ( x e. dom F -> ( A. x e. A B C_ C -> ( F ` x ) C_ C ) )
35	22, 34	sylbir 125	. . . . 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 1640	. . . 4 |- ( y e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` y ) C_ C ) )
37	3, 36	vtoclga 2619	. . 3 |- ( D e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` D ) C_ C ) )
38	37, 21	eleq2s 2132	. 2 |- ( D e. dom F -> ( A. x e. A B C_ C -> ( F ` D ) C_ C ) )
39	38	imp 115	1 |- ( ( D e. dom F /\ A. x e. A B C_ C ) -> ( F ` D ) C_ C )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   A.wral 2306   {crab 2310   _Vcvv 2557   C_ wss 2917   |-> cmpt 3818   dom cdm 4345   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fv 4910

This theorem is referenced by: (None)