Theorem fvmptss 5705

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
fvmpt2.1	|- 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		fvmpt2.1	. . . . 5 |- F = (x e. A |-> B)
2	1	dmmptss 5685	. . . 4 |- dom F C_ A
3	2	sseli 3269	. . 3 |- (D e. dom F -> D e. A)
4		fveq2 5328	. . . . . . 7 |- (y = D -> (F` y) = (F` D))
5	4	sseq1d 3298	. . . . . 6 |- (y = D -> ((F` y) C_ C <-> (F` D) C_ C))
6	5	imbi2d 307	. . . . 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 2489	. . . . . 6 |- F/_xy
8		nfra1 2664	. . . . . . 7 |- F/xA.x e. A B C_ C
9		nfmpt1 5672	. . . . . . . . . 10 |- F/_x(x e. A |-> B)
10	1, 9	nfcxfr 2486	. . . . . . . . 9 |- F/_xF
11	10, 7	nffv 5334	. . . . . . . 8 |- F/_x(F` y)
12		nfcv 2489	. . . . . . . 8 |- F/_xC
13	11, 12	nfss 3266	. . . . . . 7 |- F/x(F` y) C_ C
14	8, 13	nfim 1813	. . . . . 6 |- F/x(A.x e. A B C_ C -> (F` y) C_ C)
15		fveq2 5328	. . . . . . . 8 |- (x = y -> (F` x) = (F` y))
16	15	sseq1d 3298	. . . . . . 7 |- (x = y -> ((F` x) C_ C <-> (F` y) C_ C))
17	16	imbi2d 307	. . . . . 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 5683	. . . . . . . . . . 11 |- dom F = {x e. A | B e. _V}
19	18	rabeq2i 2856	. . . . . . . . . 10 |- (x e. dom F <-> (x e. A /\ B e. _V))
20	1	fvmpt2 5704	. . . . . . . . . . 11 |- ((x e. A /\ B e. _V) -> (F` x) = B)
21		eqimss 3323	. . . . . . . . . . 11 |- ((F` x) = B -> (F` x) C_ B)
22	20, 21	syl 15	. . . . . . . . . 10 |- ((x e. A /\ B e. _V) -> (F` x) C_ B)
23	19, 22	sylbi 187	. . . . . . . . 9 |- (x e. dom F -> (F` x) C_ B)
24		ndmfv 5349	. . . . . . . . . 10 |- (-. x e. dom F -> (F` x) = (/))
25		0ss 3579	. . . . . . . . . . 11 |- (/) C_ B
26	25	a1i 10	. . . . . . . . . 10 |- (-. x e. dom F -> (/) C_ B)
27	24, 26	eqsstrd 3305	. . . . . . . . 9 |- (-. x e. dom F -> (F` x) C_ B)
28	23, 27	pm2.61i 156	. . . . . . . 8 |- (F` x) C_ B
29		rsp 2674	. . . . . . . . 9 |- (A.x e. A B C_ C -> (x e. A -> B C_ C))
30	29	impcom 419	. . . . . . . 8 |- ((x e. A /\ A.x e. A B C_ C) -> B C_ C)
31	28, 30	syl5ss 3283	. . . . . . 7 |- ((x e. A /\ A.x e. A B C_ C) -> (F` x) C_ C)
32	31	ex 423	. . . . . 6 |- (x e. A -> (A.x e. A B C_ C -> (F` x) C_ C))
33	7, 14, 17, 32	vtoclgaf 2919	. . . . 5 |- (y e. A -> (A.x e. A B C_ C -> (F` y) C_ C))
34	6, 33	vtoclga 2920	. . . 4 |- (D e. A -> (A.x e. A B C_ C -> (F` D) C_ C))
35	34	impcom 419	. . 3 |- ((A.x e. A B C_ C /\ D e. A) -> (F` D) C_ C)
36	3, 35	sylan2 460	. 2 |- ((A.x e. A B C_ C /\ D e. dom F) -> (F` D) C_ C)
37		ndmfv 5349	. . . 4 |- (-. D e. dom F -> (F` D) = (/))
38	37	adantl 452	. . 3 |- ((A.x e. A B C_ C /\ -. D e. dom F) -> (F` D) = (/))
39		0ss 3579	. . . 4 |- (/) C_ C
40	39	a1i 10	. . 3 |- ((A.x e. A B C_ C /\ -. D e. dom F) -> (/) C_ C)
41	38, 40	eqsstrd 3305	. 2 |- ((A.x e. A B C_ C /\ -. D e. dom F) -> (F` D) C_ C)
42	36, 41	pm2.61dan 766	1 |- (A.x e. A B C_ C -> (F` D) C_ C)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2614  _Vcvv 2859   C_ wss 3257  (/)c0 3550  dom cdm 4772  ` cfv 4781   |-> cmpt 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-mpt 5652

This theorem is referenced by: (None)