Theorem fvmptssdm 5505

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 5421	. . . . . 6 ⊢ (𝑦 = 𝐷 → (𝐹‘𝑦) = (𝐹‘𝐷))
2	1	sseq1d 3126	. . . . 5 ⊢ (𝑦 = 𝐷 → ((𝐹‘𝑦) ⊆ 𝐶 ↔ (𝐹‘𝐷) ⊆ 𝐶))
3	2	imbi2d 229	. . . 4 ⊢ (𝑦 = 𝐷 → ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶)))
4		nfrab1 2610	. . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	4	nfcri 2275	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
6		nfra1 2466	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶
7		fvmpt2.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
8		nfmpt1 4021	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	7, 8	nfcxfr 2278	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
10		nfcv 2281	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
11	9, 10	nffv 5431	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
12		nfcv 2281	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
13	11, 12	nfss 3090	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) ⊆ 𝐶
14	6, 13	nfim 1551	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶)
15	5, 14	nfim 1551	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶))
16		eleq1 2202	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}))
17		fveq2 5421	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
18	17	sseq1d 3126	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ⊆ 𝐶 ↔ (𝐹‘𝑦) ⊆ 𝐶))
19	18	imbi2d 229	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶)))
20	16, 19	imbi12d 233	. . . . 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 5034	. . . . . . 7 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
22	21	eleq2i 2206	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
23	21	rabeq2i 2683	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
24	7	fvmpt2 5504	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵)
25		eqimss 3151	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = 𝐵 → (𝐹‘𝑥) ⊆ 𝐵)
26	24, 25	syl 14	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) ⊆ 𝐵)
27	23, 26	sylbi 120	. . . . . . . . 9 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) ⊆ 𝐵)
28	27	adantr 274	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝑥) ⊆ 𝐵)
29	7	dmmptss 5035	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ 𝐴
30	29	sseli 3093	. . . . . . . . 9 ⊢ (𝑥 ∈ dom 𝐹 → 𝑥 ∈ 𝐴)
31		rsp 2480	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶))
32	30, 31	mpan9 279	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
33	28, 32	sstrd 3107	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝑥) ⊆ 𝐶)
34	33	ex 114	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐹 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶))
35	22, 34	sylbir 134	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶))
36	15, 20, 35	chvar 1730	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶))
37	3, 36	vtoclga 2752	. . 3 ⊢ (𝐷 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶))
38	37, 21	eleq2s 2234	. 2 ⊢ (𝐷 ∈ dom 𝐹 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶))
39	38	imp 123	1 ⊢ ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝐷) ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2416  {crab 2420  Vcvv 2686   ⊆ wss 3071   ↦ cmpt 3989  dom cdm 4539  ‘cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fv 5131

This theorem is referenced by: (None)