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Mirrors > Home > MPE Home > Th. List > elmapssres | Structured version Visualization version GIF version |
Description: A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
elmapssres | ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8421 | . . 3 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐴:𝐶⟶𝐵) | |
2 | fssres 6537 | . . 3 ⊢ ((𝐴:𝐶⟶𝐵 ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷):𝐷⟶𝐵) | |
3 | 1, 2 | sylan 582 | . 2 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷):𝐷⟶𝐵) |
4 | elmapex 8420 | . . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) | |
5 | 4 | simpld 497 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐵 ∈ V) |
6 | 5 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → 𝐵 ∈ V) |
7 | 4 | simprd 498 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐶 ∈ V) |
8 | ssexg 5220 | . . . . 5 ⊢ ((𝐷 ⊆ 𝐶 ∧ 𝐶 ∈ V) → 𝐷 ∈ V) | |
9 | 8 | ancoms 461 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ⊆ 𝐶) → 𝐷 ∈ V) |
10 | 7, 9 | sylan 582 | . . 3 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → 𝐷 ∈ V) |
11 | 6, 10 | elmapd 8413 | . 2 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → ((𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷) ↔ (𝐴 ↾ 𝐷):𝐷⟶𝐵)) |
12 | 3, 11 | mpbird 259 | 1 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 Vcvv 3491 ⊆ wss 3929 ↾ cres 5550 ⟶wf 6344 (class class class)co 7149 ↑m cmap 8399 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 df-map 8401 |
This theorem is referenced by: nn0gsumfz 19099 mdetmul 21227 mapfzcons1cl 39391 mzpcompact2lem 39424 diophin 39445 eldiophss 39447 eldioph4b 39484 mccllem 41952 iccpartres 43652 lincresunit3lem2 44609 |
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