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Theorem elmapssres 8791

Description: A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)

**Assertion**
Ref	Expression
elmapssres	⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑_m 𝐷))

**Proof of Theorem elmapssres**
Step	Hyp	Ref	Expression
1		elmapi 8773	. . 3 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐴:𝐶⟶𝐵)
2		fssres 6689	. . 3 ⊢ ((𝐴:𝐶⟶𝐵 ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷):𝐷⟶𝐵)
3	1, 2	sylan 580	. 2 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷):𝐷⟶𝐵)
4		elmapex 8772	. . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
5	4	simpld 494	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐵 ∈ V)
6	5	adantr 480	. . 3 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → 𝐵 ∈ V)
7	4	simprd 495	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐶 ∈ V)
8		ssexg 5261	. . . . 5 ⊢ ((𝐷 ⊆ 𝐶 ∧ 𝐶 ∈ V) → 𝐷 ∈ V)
9	8	ancoms 458	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ⊆ 𝐶) → 𝐷 ∈ V)
10	7, 9	sylan 580	. . 3 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → 𝐷 ∈ V)
11	6, 10	elmapd 8764	. 2 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → ((𝐴 ↾ 𝐷) ∈ (𝐵 ↑_m 𝐷) ↔ (𝐴 ↾ 𝐷):𝐷⟶𝐵))
12	3, 11	mpbird 257	1 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑_m 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2111 Vcvv 3436 ⊆ wss 3902 ↾ cres 5618 ⟶wf 6477 (class class class)co 7346 ↑_m cmap 8750

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-map 8752

This theorem is referenced by: nn0gsumfz 19897 mdetmul 22539 elmapssresd 42280 mapfzcons1cl 42757 mzpcompact2lem 42790 diophin 42811 eldiophss 42813 eldioph4b 42850 tfsconcatrev 43387 mccllem 45643 iccpartres 47455 lincresunit3lem2 48518