elmapssres - Metamath Proof Explorer

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Theorem elmapssres 8874

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 8849	. . 3 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐴:𝐶⟶𝐵)
2		fssres 6742	. . 3 ⊢ ((𝐴:𝐶⟶𝐵 ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷):𝐷⟶𝐵)
3	1, 2	sylan 592	. 2 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷):𝐷⟶𝐵)
4		elmapex 8848	. . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
5	4	simpld 500	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐵 ∈ V)
6	5	adantr 486	. . 3 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → 𝐵 ∈ V)
7	4	simprd 501	. . . 4 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐶 ∈ V)
8		ssexg 5284	. . . . 5 ⊢ ((𝐷 ⊆ 𝐶 ∧ 𝐶 ∈ V) → 𝐷 ∈ V)
9	8	ancoms 464	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ⊆ 𝐶) → 𝐷 ∈ V)
10	7, 9	sylan 592	. . 3 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → 𝐷 ∈ V)
11	6, 10	elmapd 8840	. 2 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → ((𝐴 ↾ 𝐷) ∈ (𝐵 ↑_m 𝐷) ↔ (𝐴 ↾ 𝐷):𝐷⟶𝐵))
12	3, 11	mpbird 260	1 ⊢ ((𝐴 ∈ (𝐵 ↑_m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑_m 𝐷))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∈ wcel 2145 Vcvv 3450 ⊆ wss 3899 ↾ cres 5657 ⟶wf 6529 (class class class)co 7414 ↑_m cmap 8827

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-map 8829

This theorem is used by: elmapssresd 8875 nn0gsumfz 20112 mdetmul 22846 mapfzcons1cl 43564 mzpcompact2lem 43597 diophin 43618 eldiophss 43620 eldioph4b 43653 tfsconcatrev 44190 mccllem 46428 iccpartres 48319 lincresunit3lem2 49411

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