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Mathbox for Glauco Siliprandi |
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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 4105 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∩ 𝐵) = (𝑋 ∩ 𝐵)) | |
5 | 4 | rspceeqv 3554 | . . 3 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐴 = (𝑋 ∩ 𝐵)) → ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵)) |
6 | 1, 3, 5 | syl2anc 588 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵)) |
7 | elrestd.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑉) | |
8 | elrestd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
9 | elrest 16744 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵))) | |
10 | 7, 8, 9 | syl2anc 588 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵))) |
11 | 6, 10 | mpbird 260 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1539 ∈ wcel 2112 ∃wrex 3069 ∩ cin 3853 (class class class)co 7143 ↾t crest 16737 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-rest 16739 |
This theorem is referenced by: restuni3 42111 subsaliuncl 43349 subsalsal 43350 sssmf 43723 mbfresmf 43724 smfconst 43734 smflimlem1 43755 smfres 43773 smfco 43785 smfsuplem1 43793 |
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