Theorem fvmptss 6292

Description: If all the values of the mapping are subsets of a class C, then so is any evaluation of the mapping, even if D is not in the base set A. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypothesis

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

Assertion

Ref	Expression
fvmptss	|- ( 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 fvmptss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mptrcl.1	. . . . 5 |- F = ( x e. A |-> B )
2	1	dmmptss 5631	. . . 4 |- dom F C_ A
3	2	sseli 3599	. . 3 |- ( D e. dom F -> D e. A )
4		fveq2 6191	. . . . . . 7 |- ( y = D -> ( F ` y ) = ( F ` D ) )
5	4	sseq1d 3632	. . . . . 6 |- ( y = D -> ( ( F ` y ) C_ C <-> ( F ` D ) C_ C ) )
6	5	imbi2d 330	. . . . 5 |- ( y = D -> ( ( A. x e. A B C_ C -> ( F ` y ) C_ C ) <-> ( A. x e. A B C_ C -> ( F ` D ) C_ C ) ) )
7		nfcv 2764	. . . . . 6 |- F/_ x y
8		nfra1 2941	. . . . . . 7 |- F/ x A. x e. A B C_ C
9		nfmpt1 4747	. . . . . . . . . 10 |- F/_ x ( x e. A |-> B )
10	1, 9	nfcxfr 2762	. . . . . . . . 9 |- F/_ x F
11	10, 7	nffv 6198	. . . . . . . 8 |- F/_ x ( F ` y )
12		nfcv 2764	. . . . . . . 8 |- F/_ x C
13	11, 12	nfss 3596	. . . . . . 7 |- F/ x ( F ` y ) C_ C
14	8, 13	nfim 1825	. . . . . 6 |- F/ x ( A. x e. A B C_ C -> ( F ` y ) C_ C )
15		fveq2 6191	. . . . . . . 8 |- ( x = y -> ( F ` x ) = ( F ` y ) )
16	15	sseq1d 3632	. . . . . . 7 |- ( x = y -> ( ( F ` x ) C_ C <-> ( F ` y ) C_ C ) )
17	16	imbi2d 330	. . . . . 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 ) ) )
18	1	dmmpt 5630	. . . . . . . . . . 11 |- dom F = { x e. A | B e. _V }
19	18	rabeq2i 3197	. . . . . . . . . 10 |- ( x e. dom F <-> ( x e. A /\ B e. _V ) )
20	1	fvmpt2 6291	. . . . . . . . . . 11 |- ( ( x e. A /\ B e. _V ) -> ( F ` x ) = B )
21		eqimss 3657	. . . . . . . . . . 11 |- ( ( F ` x ) = B -> ( F ` x ) C_ B )
22	20, 21	syl 17	. . . . . . . . . 10 |- ( ( x e. A /\ B e. _V ) -> ( F ` x ) C_ B )
23	19, 22	sylbi 207	. . . . . . . . 9 |- ( x e. dom F -> ( F ` x ) C_ B )
24		ndmfv 6218	. . . . . . . . . 10 |- ( -. x e. dom F -> ( F ` x ) = (/) )
25		0ss 3972	. . . . . . . . . 10 |- (/) C_ B
26	24, 25	syl6eqss 3655	. . . . . . . . 9 |- ( -. x e. dom F -> ( F ` x ) C_ B )
27	23, 26	pm2.61i 176	. . . . . . . 8 |- ( F ` x ) C_ B
28		rsp 2929	. . . . . . . . 9 |- ( A. x e. A B C_ C -> ( x e. A -> B C_ C ) )
29	28	impcom 446	. . . . . . . 8 |- ( ( x e. A /\ A. x e. A B C_ C ) -> B C_ C )
30	27, 29	syl5ss 3614	. . . . . . 7 |- ( ( x e. A /\ A. x e. A B C_ C ) -> ( F ` x ) C_ C )
31	30	ex 450	. . . . . 6 |- ( x e. A -> ( A. x e. A B C_ C -> ( F ` x ) C_ C ) )
32	7, 14, 17, 31	vtoclgaf 3271	. . . . 5 |- ( y e. A -> ( A. x e. A B C_ C -> ( F ` y ) C_ C ) )
33	6, 32	vtoclga 3272	. . . 4 |- ( D e. A -> ( A. x e. A B C_ C -> ( F ` D ) C_ C ) )
34	33	impcom 446	. . 3 |- ( ( A. x e. A B C_ C /\ D e. A ) -> ( F ` D ) C_ C )
35	3, 34	sylan2 491	. 2 |- ( ( A. x e. A B C_ C /\ D e. dom F ) -> ( F ` D ) C_ C )
36		ndmfv 6218	. . . 4 |- ( -. D e. dom F -> ( F ` D ) = (/) )
37	36	adantl 482	. . 3 |- ( ( A. x e. A B C_ C /\ -. D e. dom F ) -> ( F ` D ) = (/) )
38		0ss 3972	. . 3 |- (/) C_ C
39	37, 38	syl6eqss 3655	. 2 |- ( ( A. x e. A B C_ C /\ -. D e. dom F ) -> ( F ` D ) C_ C )
40	35, 39	pm2.61dan 832	1 |- ( A. x e. A B C_ C -> ( F ` D ) C_ C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  relmptopab  6883  ovmptss  7258