Theorem fvmptssdm 5330

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 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
fvmptssdm	⊢ ((𝐷 ∈ dom 𝐹 ∧ ∀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 fvmptssdm

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5251	. . . . . 6 ⊢ (𝑦 = 𝐷 → (𝐹‘𝑦) = (𝐹‘𝐷))
2	1	sseq1d 3037	. . . . 5 ⊢ (𝑦 = 𝐷 → ((𝐹‘𝑦) ⊆ 𝐶 ↔ (𝐹‘𝐷) ⊆ 𝐶))
3	2	imbi2d 228	. . . 4 ⊢ (𝑦 = 𝐷 → ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶)))
4		nfrab1 2539	. . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	4	nfcri 2217	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
6		nfra1 2403	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶
7		fvmpt2.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
8		nfmpt1 3897	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	7, 8	nfcxfr 2220	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
10		nfcv 2223	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
11	9, 10	nffv 5258	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
12		nfcv 2223	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
13	11, 12	nfss 3003	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) ⊆ 𝐶
14	6, 13	nfim 1505	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶)
15	5, 14	nfim 1505	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶))
16		eleq1 2145	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}))
17		fveq2 5251	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
18	17	sseq1d 3037	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ⊆ 𝐶 ↔ (𝐹‘𝑦) ⊆ 𝐶))
19	18	imbi2d 228	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶)))
20	16, 19	imbi12d 232	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶)) ↔ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶))))
21	7	dmmpt 4878	. . . . . . 7 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
22	21	eleq2i 2149	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
23	21	rabeq2i 2609	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
24	7	fvmpt2 5329	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵)
25		eqimss 3062	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = 𝐵 → (𝐹‘𝑥) ⊆ 𝐵)
26	24, 25	syl 14	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) ⊆ 𝐵)
27	23, 26	sylbi 119	. . . . . . . . 9 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) ⊆ 𝐵)
28	27	adantr 270	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝑥) ⊆ 𝐵)
29	7	dmmptss 4879	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ 𝐴
30	29	sseli 3006	. . . . . . . . 9 ⊢ (𝑥 ∈ dom 𝐹 → 𝑥 ∈ 𝐴)
31		rsp 2417	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶))
32	30, 31	mpan9 275	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
33	28, 32	sstrd 3020	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝑥) ⊆ 𝐶)
34	33	ex 113	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐹 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶))
35	22, 34	sylbir 133	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶))
36	15, 20, 35	chvar 1682	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶))
37	3, 36	vtoclga 2675	. . 3 ⊢ (𝐷 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶))
38	37, 21	eleq2s 2177	. 2 ⊢ (𝐷 ∈ dom 𝐹 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶))
39	38	imp 122	1 ⊢ ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ∈ wcel 1434  ∀wral 2353  {crab 2357  Vcvv 2612   ⊆ wss 2984   ↦ cmpt 3865  dom cdm 4399  ‘cfv 4967

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fv 4975

This theorem is referenced by: (None)