Theorem fvmptss 7004

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
mptrcl.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		mptrcl.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	dmmptss 6244	. . . 4 ⊢ dom 𝐹 ⊆ 𝐴
3	2	sseli 3934	. . 3 ⊢ (𝐷 ∈ dom 𝐹 → 𝐷 ∈ 𝐴)
4		fveq2 6883	. . . . . . 7 ⊢ (𝑦 = 𝐷 → (𝐹‘𝑦) = (𝐹‘𝐷))
5	4	sseq1d 3969	. . . . . 6 ⊢ (𝑦 = 𝐷 → ((𝐹‘𝑦) ⊆ 𝐶 ↔ (𝐹‘𝐷) ⊆ 𝐶))
6	5	imbi2d 343	. . . . 5 ⊢ (𝑦 = 𝐷 → ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶)))
7		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑥𝑦
8		nfra1 3289	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶
9		nfmpt1 5211	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
10	1, 9	nfcxfr 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
11	10, 7	nffv 6893	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
12		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
13	11, 12	nfss 3931	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) ⊆ 𝐶
14	8, 13	nfim 1926	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶)
15		fveq2 6883	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
16	15	sseq1d 3969	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ⊆ 𝐶 ↔ (𝐹‘𝑦) ⊆ 𝐶))
17	16	imbi2d 343	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶)))
18	1	dmmpt 6243	. . . . . . . . . . 11 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
19	18	reqabi 3439	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
20	1	fvmpt2 7003	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵)
21		eqimss 3996	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = 𝐵 → (𝐹‘𝑥) ⊆ 𝐵)
22	20, 21	syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) ⊆ 𝐵)
23	19, 22	sylbi 220	. . . . . . . . 9 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) ⊆ 𝐵)
24		ndmfv 6915	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = ∅)
25		0ss 4358	. . . . . . . . . 10 ⊢ ∅ ⊆ 𝐵
26	24, 25	eqsstrdi 3982	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) ⊆ 𝐵)
27	23, 26	pm2.61i 184	. . . . . . . 8 ⊢ (𝐹‘𝑥) ⊆ 𝐵
28		rsp 3253	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶))
29	28	impcom 412	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
30	27, 29	sstrid 3949	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (𝐹‘𝑥) ⊆ 𝐶)
31	30	ex 417	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑥) ⊆ 𝐶))
32	7, 14, 17, 31	vtoclgaf 3541	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝑦) ⊆ 𝐶))
33	6, 32	vtoclga 3542	. . . 4 ⊢ (𝐷 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶))
34	33	impcom 412	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ 𝐷 ∈ 𝐴) → (𝐹‘𝐷) ⊆ 𝐶)
35	3, 34	sylan2 604	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ 𝐷 ∈ dom 𝐹) → (𝐹‘𝐷) ⊆ 𝐶)
36		ndmfv 6915	. . . 4 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) = ∅)
37	36	adantl 486	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹‘𝐷) = ∅)
38		0ss 4358	. . 3 ⊢ ∅ ⊆ 𝐶
39	37, 38	eqsstrdi 3982	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹‘𝐷) ⊆ 𝐶)
40	35, 39	pm2.61dan 824	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ⊆ wss 3906  ∅c0 4287   ↦ cmpt 5193  dom cdm 5663  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is referenced by:  relmptopab  7662  ovmptss  8089