Theorem fvmptss 5706

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 ∈ A ↦ B)

Assertion

Ref	Expression
fvmptss	⊢ (∀x ∈ A B ⊆ C → (F ‘D) ⊆ 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 ∈ A ↦ B)
2	1	dmmptss 5686	. . . 4 ⊢ dom F ⊆ A
3	2	sseli 3270	. . 3 ⊢ (D ∈ dom F → D ∈ A)
4		fveq2 5329	. . . . . . 7 ⊢ (y = D → (F ‘y) = (F ‘D))
5	4	sseq1d 3299	. . . . . 6 ⊢ (y = D → ((F ‘y) ⊆ C ↔ (F ‘D) ⊆ C))
6	5	imbi2d 307	. . . . 5 ⊢ (y = D → ((∀x ∈ A B ⊆ C → (F ‘y) ⊆ C) ↔ (∀x ∈ A B ⊆ C → (F ‘D) ⊆ C)))
7		nfcv 2490	. . . . . 6 ⊢ Ⅎxy
8		nfra1 2665	. . . . . . 7 ⊢ Ⅎx∀x ∈ A B ⊆ C
9		nfmpt1 5673	. . . . . . . . . 10 ⊢ Ⅎx(x ∈ A ↦ B)
10	1, 9	nfcxfr 2487	. . . . . . . . 9 ⊢ ℲxF
11	10, 7	nffv 5335	. . . . . . . 8 ⊢ Ⅎx(F ‘y)
12		nfcv 2490	. . . . . . . 8 ⊢ ℲxC
13	11, 12	nfss 3267	. . . . . . 7 ⊢ Ⅎx(F ‘y) ⊆ C
14	8, 13	nfim 1813	. . . . . 6 ⊢ Ⅎx(∀x ∈ A B ⊆ C → (F ‘y) ⊆ C)
15		fveq2 5329	. . . . . . . 8 ⊢ (x = y → (F ‘x) = (F ‘y))
16	15	sseq1d 3299	. . . . . . 7 ⊢ (x = y → ((F ‘x) ⊆ C ↔ (F ‘y) ⊆ C))
17	16	imbi2d 307	. . . . . 6 ⊢ (x = y → ((∀x ∈ A B ⊆ C → (F ‘x) ⊆ C) ↔ (∀x ∈ A B ⊆ C → (F ‘y) ⊆ C)))
18	1	dmmpt 5684	. . . . . . . . . . 11 ⊢ dom F = {x ∈ A ∣ B ∈ V}
19	18	rabeq2i 2857	. . . . . . . . . 10 ⊢ (x ∈ dom F ↔ (x ∈ A ∧ B ∈ V))
20	1	fvmpt2 5705	. . . . . . . . . . 11 ⊢ ((x ∈ A ∧ B ∈ V) → (F ‘x) = B)
21		eqimss 3324	. . . . . . . . . . 11 ⊢ ((F ‘x) = B → (F ‘x) ⊆ B)
22	20, 21	syl 15	. . . . . . . . . 10 ⊢ ((x ∈ A ∧ B ∈ V) → (F ‘x) ⊆ B)
23	19, 22	sylbi 187	. . . . . . . . 9 ⊢ (x ∈ dom F → (F ‘x) ⊆ B)
24		ndmfv 5350	. . . . . . . . . 10 ⊢ (¬ x ∈ dom F → (F ‘x) = ∅)
25		0ss 3580	. . . . . . . . . . 11 ⊢ ∅ ⊆ B
26	25	a1i 10	. . . . . . . . . 10 ⊢ (¬ x ∈ dom F → ∅ ⊆ B)
27	24, 26	eqsstrd 3306	. . . . . . . . 9 ⊢ (¬ x ∈ dom F → (F ‘x) ⊆ B)
28	23, 27	pm2.61i 156	. . . . . . . 8 ⊢ (F ‘x) ⊆ B
29		rsp 2675	. . . . . . . . 9 ⊢ (∀x ∈ A B ⊆ C → (x ∈ A → B ⊆ C))
30	29	impcom 419	. . . . . . . 8 ⊢ ((x ∈ A ∧ ∀x ∈ A B ⊆ C) → B ⊆ C)
31	28, 30	syl5ss 3284	. . . . . . 7 ⊢ ((x ∈ A ∧ ∀x ∈ A B ⊆ C) → (F ‘x) ⊆ C)
32	31	ex 423	. . . . . 6 ⊢ (x ∈ A → (∀x ∈ A B ⊆ C → (F ‘x) ⊆ C))
33	7, 14, 17, 32	vtoclgaf 2920	. . . . 5 ⊢ (y ∈ A → (∀x ∈ A B ⊆ C → (F ‘y) ⊆ C))
34	6, 33	vtoclga 2921	. . . 4 ⊢ (D ∈ A → (∀x ∈ A B ⊆ C → (F ‘D) ⊆ C))
35	34	impcom 419	. . 3 ⊢ ((∀x ∈ A B ⊆ C ∧ D ∈ A) → (F ‘D) ⊆ C)
36	3, 35	sylan2 460	. 2 ⊢ ((∀x ∈ A B ⊆ C ∧ D ∈ dom F) → (F ‘D) ⊆ C)
37		ndmfv 5350	. . . 4 ⊢ (¬ D ∈ dom F → (F ‘D) = ∅)
38	37	adantl 452	. . 3 ⊢ ((∀x ∈ A B ⊆ C ∧ ¬ D ∈ dom F) → (F ‘D) = ∅)
39		0ss 3580	. . . 4 ⊢ ∅ ⊆ C
40	39	a1i 10	. . 3 ⊢ ((∀x ∈ A B ⊆ C ∧ ¬ D ∈ dom F) → ∅ ⊆ C)
41	38, 40	eqsstrd 3306	. 2 ⊢ ((∀x ∈ A B ⊆ C ∧ ¬ D ∈ dom F) → (F ‘D) ⊆ C)
42	36, 41	pm2.61dan 766	1 ⊢ (∀x ∈ A B ⊆ C → (F ‘D) ⊆ C)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  Vcvv 2860   ⊆ wss 3258  ∅c0 3551  dom cdm 4773   ‘cfv 4782   ↦ cmpt 5652

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fv 4796  df-mpt 5653

This theorem is referenced by: (None)