Theorem fvmptssdm 5255

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 5178	. . . . . 6 ⊢ (𝑦 = 𝐷 → (𝐹‘𝑦) = (𝐹‘𝐷))
2	1	sseq1d 2972	. . . . 5 ⊢ (𝑦 = 𝐷 → ((𝐹‘𝑦) ⊆ 𝐶 ↔ (𝐹‘𝐷) ⊆ 𝐶))
3	2	imbi2d 219	. . . 4 ⊢ (𝑦 = 𝐷 → ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶)))
4		nfrab1 2489	. . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	4	nfcri 2172	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
6		nfra1 2355	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶
7		fvmpt2.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
8		nfmpt1 3850	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	7, 8	nfcxfr 2175	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
10		nfcv 2178	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
11	9, 10	nffv 5185	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
12		nfcv 2178	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
13	11, 12	nfss 2938	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) ⊆ 𝐶
14	6, 13	nfim 1464	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶)
15	5, 14	nfim 1464	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶))
16		eleq1 2100	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}))
17		fveq2 5178	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
18	17	sseq1d 2972	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ⊆ 𝐶 ↔ (𝐹‘𝑦) ⊆ 𝐶))
19	18	imbi2d 219	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶)))
20	16, 19	imbi12d 223	. . . . 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 4816	. . . . . . 7 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
22	21	eleq2i 2104	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
23	21	rabeq2i 2554	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
24	7	fvmpt2 5254	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵)
25		eqimss 2997	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = 𝐵 → (𝐹‘𝑥) ⊆ 𝐵)
26	24, 25	syl 14	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) ⊆ 𝐵)
27	23, 26	sylbi 114	. . . . . . . . 9 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) ⊆ 𝐵)
28	27	adantr 261	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝑥) ⊆ 𝐵)
29	7	dmmptss 4817	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ 𝐴
30	29	sseli 2941	. . . . . . . . 9 ⊢ (𝑥 ∈ dom 𝐹 → 𝑥 ∈ 𝐴)
31		rsp 2369	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶))
32	30, 31	mpan9 265	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
33	28, 32	sstrd 2955	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝑥) ⊆ 𝐶)
34	33	ex 108	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐹 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶))
35	22, 34	sylbir 125	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶))
36	15, 20, 35	chvar 1640	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶))
37	3, 36	vtoclga 2619	. . 3 ⊢ (𝐷 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶))
38	37, 21	eleq2s 2132	. 2 ⊢ (𝐷 ∈ dom 𝐹 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶))
39	38	imp 115	1 ⊢ ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310  Vcvv 2557   ⊆ wss 2917   ↦ cmpt 3818  dom cdm 4345  ‘cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fv 4910

This theorem is referenced by: (None)