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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrestd | Structured version Visualization version GIF version |
Description: A sufficient condition for being an open set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
elrestd.1 | ⊢ (𝜑 → 𝐽 ∈ 𝑉) |
elrestd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
elrestd.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐽) |
elrestd.4 | ⊢ 𝐴 = (𝑋 ∩ 𝐵) |
Ref | Expression |
---|---|
elrestd | ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrestd.3 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐽) | |
2 | elrestd.4 | . . . 4 ⊢ 𝐴 = (𝑋 ∩ 𝐵) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐴 = (𝑋 ∩ 𝐵)) |
4 | ineq1 4178 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∩ 𝐵) = (𝑋 ∩ 𝐵)) | |
5 | 4 | rspceeqv 3635 | . . 3 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐴 = (𝑋 ∩ 𝐵)) → ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵)) |
6 | 1, 3, 5 | syl2anc 584 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵)) |
7 | elrestd.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑉) | |
8 | elrestd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
9 | elrest 16689 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵))) | |
10 | 7, 8, 9 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵))) |
11 | 6, 10 | mpbird 258 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ∩ cin 3932 (class class class)co 7145 ↾t crest 16682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 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-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-rest 16684 |
This theorem is referenced by: restuni3 41261 subsaliuncl 42518 subsalsal 42519 sssmf 42892 mbfresmf 42893 smfconst 42903 smflimlem1 42924 smfres 42942 smfco 42954 smfsuplem1 42962 |
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