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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 8868 | . . 3 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐴:𝐶⟶𝐵) | |
2 | fssres 6763 | . . 3 ⊢ ((𝐴:𝐶⟶𝐵 ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷):𝐷⟶𝐵) | |
3 | 1, 2 | sylan 578 | . 2 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷):𝐷⟶𝐵) |
4 | elmapex 8867 | . . . . 5 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) | |
5 | 4 | simpld 493 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐵 ∈ V) |
6 | 5 | adantr 479 | . . 3 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → 𝐵 ∈ V) |
7 | 4 | simprd 494 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐶 ∈ V) |
8 | ssexg 5324 | . . . . 5 ⊢ ((𝐷 ⊆ 𝐶 ∧ 𝐶 ∈ V) → 𝐷 ∈ V) | |
9 | 8 | ancoms 457 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ⊆ 𝐶) → 𝐷 ∈ V) |
10 | 7, 9 | sylan 578 | . . 3 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → 𝐷 ∈ V) |
11 | 6, 10 | elmapd 8859 | . 2 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → ((𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷) ↔ (𝐴 ↾ 𝐷):𝐷⟶𝐵)) |
12 | 3, 11 | mpbird 256 | 1 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 Vcvv 3461 ⊆ wss 3944 ↾ cres 5680 ⟶wf 6545 (class class class)co 7419 ↑m cmap 8845 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-map 8847 |
This theorem is referenced by: nn0gsumfz 19951 mdetmul 22569 elmapssresd 41865 mapfzcons1cl 42280 mzpcompact2lem 42313 diophin 42334 eldiophss 42336 eldioph4b 42373 tfsconcatrev 42919 mccllem 45123 iccpartres 46895 lincresunit3lem2 47734 |
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