Theorem fvmptssdm 5621

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 5534	. . . . . 6 ⊢ (𝑦 = 𝐷 → (𝐹‘𝑦) = (𝐹‘𝐷))
2	1	sseq1d 3199	. . . . 5 ⊢ (𝑦 = 𝐷 → ((𝐹‘𝑦) ⊆ 𝐶 ↔ (𝐹‘𝐷) ⊆ 𝐶))
3	2	imbi2d 230	. . . 4 ⊢ (𝑦 = 𝐷 → ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶)))
4		nfrab1 2670	. . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	4	nfcri 2326	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
6		nfra1 2521	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶
7		fvmpt2.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
8		nfmpt1 4111	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	7, 8	nfcxfr 2329	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
10		nfcv 2332	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
11	9, 10	nffv 5544	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
12		nfcv 2332	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
13	11, 12	nfss 3163	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) ⊆ 𝐶
14	6, 13	nfim 1583	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶)
15	5, 14	nfim 1583	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶))
16		eleq1 2252	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}))
17		fveq2 5534	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
18	17	sseq1d 3199	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ⊆ 𝐶 ↔ (𝐹‘𝑦) ⊆ 𝐶))
19	18	imbi2d 230	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶)))
20	16, 19	imbi12d 234	. . . . 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 5142	. . . . . . 7 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
22	21	eleq2i 2256	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
23	21	rabeq2i 2749	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
24	7	fvmpt2 5620	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵)
25		eqimss 3224	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = 𝐵 → (𝐹‘𝑥) ⊆ 𝐵)
26	24, 25	syl 14	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) ⊆ 𝐵)
27	23, 26	sylbi 121	. . . . . . . . 9 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) ⊆ 𝐵)
28	27	adantr 276	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝑥) ⊆ 𝐵)
29	7	dmmptss 5143	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ 𝐴
30	29	sseli 3166	. . . . . . . . 9 ⊢ (𝑥 ∈ dom 𝐹 → 𝑥 ∈ 𝐴)
31		rsp 2537	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶))
32	30, 31	mpan9 281	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
33	28, 32	sstrd 3180	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝑥) ⊆ 𝐶)
34	33	ex 115	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐹 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶))
35	22, 34	sylbir 135	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶))
36	15, 20, 35	chvar 1768	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶))
37	3, 36	vtoclga 2818	. . 3 ⊢ (𝐷 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶))
38	37, 21	eleq2s 2284	. 2 ⊢ (𝐷 ∈ dom 𝐹 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶))
39	38	imp 124	1 ⊢ ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  ∀wral 2468  {crab 2472  Vcvv 2752   ⊆ wss 3144   ↦ cmpt 4079  dom cdm 4644  ‘cfv 5235

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fv 5243

This theorem is referenced by: (None)