Theorem fvmptssdm 5727

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 5635	. . . . . 6 |- ( y = D -> ( F ` y ) = ( F ` D ) )
2	1	sseq1d 3254	. . . . 5 |- ( y = D -> ( ( F ` y ) C_ C <-> ( F ` D ) C_ C ) )
3	2	imbi2d 230	. . . 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 2711	. . . . . . 7 |- F/_ x { x e. A | B e. _V }
5	4	nfcri 2366	. . . . . 6 |- F/ x y e. { x e. A | B e. _V }
6		nfra1 2561	. . . . . . 7 |- F/ x A. x e. A B C_ C
7		fvmpt2.1	. . . . . . . . . 10 |- F = ( x e. A |-> B )
8		nfmpt1 4180	. . . . . . . . . 10 |- F/_ x ( x e. A |-> B )
9	7, 8	nfcxfr 2369	. . . . . . . . 9 |- F/_ x F
10		nfcv 2372	. . . . . . . . 9 |- F/_ x y
11	9, 10	nffv 5645	. . . . . . . 8 |- F/_ x ( F ` y )
12		nfcv 2372	. . . . . . . 8 |- F/_ x C
13	11, 12	nfss 3218	. . . . . . 7 |- F/ x ( F ` y ) C_ C
14	6, 13	nfim 1618	. . . . . 6 |- F/ x ( A. x e. A B C_ C -> ( F ` y ) C_ C )
15	5, 14	nfim 1618	. . . . 5 |- F/ x ( y e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` y ) C_ C ) )
16		eleq1 2292	. . . . . 6 |- ( x = y -> ( x e. { x e. A | B e. _V } <-> y e. { x e. A | B e. _V } ) )
17		fveq2 5635	. . . . . . . 8 |- ( x = y -> ( F ` x ) = ( F ` y ) )
18	17	sseq1d 3254	. . . . . . 7 |- ( x = y -> ( ( F ` x ) C_ C <-> ( F ` y ) C_ C ) )
19	18	imbi2d 230	. . . . . 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 234	. . . . 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 5230	. . . . . . 7 |- dom F = { x e. A | B e. _V }
22	21	eleq2i 2296	. . . . . 6 |- ( x e. dom F <-> x e. { x e. A | B e. _V } )
23	21	rabeq2i 2797	. . . . . . . . . 10 |- ( x e. dom F <-> ( x e. A /\ B e. _V ) )
24	7	fvmpt2 5726	. . . . . . . . . . 11 |- ( ( x e. A /\ B e. _V ) -> ( F ` x ) = B )
25		eqimss 3279	. . . . . . . . . . 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 121	. . . . . . . . 9 |- ( x e. dom F -> ( F ` x ) C_ B )
28	27	adantr 276	. . . . . . . 8 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> ( F ` x ) C_ B )
29	7	dmmptss 5231	. . . . . . . . . 10 |- dom F C_ A
30	29	sseli 3221	. . . . . . . . 9 |- ( x e. dom F -> x e. A )
31		rsp 2577	. . . . . . . . 9 |- ( A. x e. A B C_ C -> ( x e. A -> B C_ C ) )
32	30, 31	mpan9 281	. . . . . . . 8 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> B C_ C )
33	28, 32	sstrd 3235	. . . . . . 7 |- ( ( x e. dom F /\ A. x e. A B C_ C ) -> ( F ` x ) C_ C )
34	33	ex 115	. . . . . 6 |- ( x e. dom F -> ( A. x e. A B C_ C -> ( F ` x ) C_ C ) )
35	22, 34	sylbir 135	. . . . 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 1803	. . . 4 |- ( y e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` y ) C_ C ) )
37	3, 36	vtoclga 2868	. . 3 |- ( D e. { x e. A | B e. _V } -> ( A. x e. A B C_ C -> ( F ` D ) C_ C ) )
38	37, 21	eleq2s 2324	. 2 |- ( D e. dom F -> ( A. x e. A B C_ C -> ( F ` D ) C_ C ) )
39	38	imp 124	1 |- ( ( D e. dom F /\ A. x e. A B C_ C ) -> ( F ` D ) C_ C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   {crab 2512   _Vcvv 2800   C_ wss 3198   |-> cmpt 4148   dom cdm 4723   ` cfv 5324

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fv 5332

This theorem is referenced by:  relmptopab  6219