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Mirrors > Home > MPE Home > Th. List > Mathboxes > restuni5 | Structured version Visualization version GIF version |
Description: The underlying set of a subspace induced by the ↾t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
restuni5.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
restuni5 | ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ 𝑉) | |
2 | id 22 | . . . . 5 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋) | |
3 | restuni5.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 2, 3 | sseqtrdi 3937 | . . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ ∪ 𝐽) |
5 | 4 | adantl 485 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ 𝐽) |
6 | 1, 5 | restuni4 42284 | . 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) = 𝐴) |
7 | 6 | eqcomd 2742 | 1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 ∪ cuni 4805 (class class class)co 7191 ↾t crest 16879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-rest 16881 |
This theorem is referenced by: (None) |
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