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Mirrors > Home > MPE Home > Th. List > Mathboxes > restsubel | Structured version Visualization version GIF version |
Description: A subset belongs in the space it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
Ref | Expression |
---|---|
restsubel.1 | ⊢ (𝜑 → 𝐽 ∈ 𝑉) |
restsubel.2 | ⊢ (𝜑 → ∪ 𝐽 ∈ 𝐽) |
restsubel.3 | ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽) |
Ref | Expression |
---|---|
restsubel | ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restsubel.2 | . . 3 ⊢ (𝜑 → ∪ 𝐽 ∈ 𝐽) | |
2 | ineq1 4205 | . . . . 5 ⊢ (𝑥 = ∪ 𝐽 → (𝑥 ∩ 𝐴) = (∪ 𝐽 ∩ 𝐴)) | |
3 | 2 | eqeq2d 2743 | . . . 4 ⊢ (𝑥 = ∪ 𝐽 → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴))) |
4 | 3 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = ∪ 𝐽) → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴))) |
5 | incom 4201 | . . . . . 6 ⊢ (∪ 𝐽 ∩ 𝐴) = (𝐴 ∩ ∪ 𝐽) | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = (𝐴 ∩ ∪ 𝐽)) |
7 | restsubel.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽) | |
8 | df-ss 3965 | . . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) = 𝐴) | |
9 | 7, 8 | sylib 217 | . . . . 5 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝐽) = 𝐴) |
10 | 6, 9 | eqtrd 2772 | . . . 4 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = 𝐴) |
11 | 10 | eqcomd 2738 | . . 3 ⊢ (𝜑 → 𝐴 = (∪ 𝐽 ∩ 𝐴)) |
12 | 1, 4, 11 | rspcedvd 3614 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴)) |
13 | restsubel.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑉) | |
14 | 1, 7 | ssexd 5324 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
15 | elrest 17375 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴))) | |
16 | 13, 14, 15 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴))) |
17 | 12, 16 | mpbird 256 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 Vcvv 3474 ∩ cin 3947 ⊆ wss 3948 ∪ cuni 4908 (class class class)co 7411 ↾t crest 17368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-rest 17370 |
This theorem is referenced by: toprestsubel 43930 |
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