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Theorem elmapssres 8801

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 8783	. . 3 ⊢ (𝐴 ∈ (𝐵 ↑_m 𝐶) → 𝐴:𝐶⟶𝐵)
2		fssres 6694	. . 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 8782	. . . . 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 5265	. . . . 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 8774	. 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 2109 Vcvv 3438 ⊆ wss 3905 ↾ cres 5625 ⟶wf 6482 (class class class)co 7353 ↑_m cmap 8760

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-fv 6494 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7931 df-2nd 7932 df-map 8762

This theorem is referenced by: nn0gsumfz 19881 mdetmul 22526 elmapssresd 42214 mapfzcons1cl 42691 mzpcompact2lem 42724 diophin 42745 eldiophss 42747 eldioph4b 42784 tfsconcatrev 43321 mccllem 45579 iccpartres 47403 lincresunit3lem2 48466