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Theorem elmapssres 8863

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 8845	. . 3 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐴:𝐶⟶𝐵)
2		fssres 6751	. . 3 ⊢ ((𝐴:𝐶⟶𝐵 ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷):𝐷⟶𝐵)
3	1, 2	sylan 579	. 2 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷):𝐷⟶𝐵)
4		elmapex 8844	. . . . 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 5316	. . . . 5 ⊢ ((𝐷 ⊆ 𝐶 ∧ 𝐶 ∈ V) → 𝐷 ∈ V)
9	8	ancoms 458	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ⊆ 𝐶) → 𝐷 ∈ V)
10	7, 9	sylan 579	. . 3 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → 𝐷 ∈ V)
11	6, 10	elmapd 8836	. 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 2098 Vcvv 3468 ⊆ wss 3943 ↾ cres 5671 ⟶wf 6533 (class class class)co 7405 ↑_m cmap 8822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-map 8824

This theorem is referenced by: nn0gsumfz 19904 mdetmul 22480 elmapssresd 41632 mapfzcons1cl 42039 mzpcompact2lem 42072 diophin 42093 eldiophss 42095 eldioph4b 42132 tfsconcatrev 42679 mccllem 44890 iccpartres 46663 lincresunit3lem2 47441