elmapssres - Metamath Proof Explorer

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Theorem elmapssres 8878

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 8853	. . 3 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐴:𝐶⟶𝐵)
2		fssres 6740	. . 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 8852	. . . . 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 5281	. . . . 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 8844	. 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 3451 ⊆ wss 3899 ↾ cres 5653 ⟶wf 6527 (class class class)co 7412 ↑_m cmap 8831

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 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 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 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-fv 6539 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7990 df-2nd 7991 df-map 8833

This theorem is used by: elmapssresd 8879 nn0gsumfz 20178 mdetmul 22918 mapfzcons1cl 43682 mzpcompact2lem 43715 diophin 43736 eldiophss 43738 eldioph4b 43771 tfsconcatrev 44308 mccllem 46553 iccpartres 48444 lincresunit3lem2 49536

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