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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 4206 | . . . . 5 ⊢ (𝑥 = ∪ 𝐽 → (𝑥 ∩ 𝐴) = (∪ 𝐽 ∩ 𝐴)) | |
3 | 2 | eqeq2d 2744 | . . . 4 ⊢ (𝑥 = ∪ 𝐽 → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴))) |
4 | 3 | adantl 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = ∪ 𝐽) → (𝐴 = (𝑥 ∩ 𝐴) ↔ 𝐴 = (∪ 𝐽 ∩ 𝐴))) |
5 | incom 4202 | . . . . . 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 3966 | . . . . . 6 ⊢ (𝐴 ⊆ ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) = 𝐴) | |
9 | 7, 8 | sylib 217 | . . . . 5 ⊢ (𝜑 → (𝐴 ∩ ∪ 𝐽) = 𝐴) |
10 | 6, 9 | eqtrd 2773 | . . . 4 ⊢ (𝜑 → (∪ 𝐽 ∩ 𝐴) = 𝐴) |
11 | 10 | eqcomd 2739 | . . 3 ⊢ (𝜑 → 𝐴 = (∪ 𝐽 ∩ 𝐴)) |
12 | 1, 4, 11 | rspcedvd 3615 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴)) |
13 | restsubel.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑉) | |
14 | 1, 7 | ssexd 5325 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
15 | elrest 17373 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴))) | |
16 | 13, 14, 15 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐴))) |
17 | 12, 16 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 ∪ cuni 4909 (class class class)co 7409 ↾t crest 17366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-rest 17368 |
This theorem is referenced by: toprestsubel 43848 |
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