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Mirrors > Home > MPE Home > Th. List > Mathboxes > restuni4 | 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 |
---|---|
restuni4.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
restuni4.2 | ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴) |
Ref | Expression |
---|---|
restuni4 | ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4175 | . . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)) |
3 | restuni4.2 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴) | |
4 | dfss 3950 | . . 3 ⊢ (𝐵 ⊆ ∪ 𝐴 ↔ 𝐵 = (𝐵 ∩ ∪ 𝐴)) | |
5 | 3, 4 | sylib 219 | . 2 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ ∪ 𝐴)) |
6 | restuni4.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | 6 | uniexd 7457 | . . . 4 ⊢ (𝜑 → ∪ 𝐴 ∈ V) |
8 | 7, 3 | ssexd 5219 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
9 | 6, 8 | restuni3 41261 | . 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) |
10 | 2, 5, 9 | 3eqtr4rd 2864 | 1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 ∪ cuni 4830 (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: restuni6 41265 restuni5 41266 subsaluni 42520 issmflelem 42898 smfpimltxr 42901 issmfgtlem 42909 issmfgt 42910 issmfgelem 42922 smfpimgtxr 42933 smfresal 42940 |
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