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Mathbox for Glauco Siliprandi |
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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 4128 | . . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)) |
3 | restuni4.2 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴) | |
4 | dfss 3899 | . . 3 ⊢ (𝐵 ⊆ ∪ 𝐴 ↔ 𝐵 = (𝐵 ∩ ∪ 𝐴)) | |
5 | 3, 4 | sylib 221 | . 2 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ ∪ 𝐴)) |
6 | restuni4.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | 6 | uniexd 7448 | . . . 4 ⊢ (𝜑 → ∪ 𝐴 ∈ V) |
8 | 7, 3 | ssexd 5192 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
9 | 6, 8 | restuni3 41753 | . 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) |
10 | 2, 5, 9 | 3eqtr4rd 2844 | 1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∪ cuni 4800 (class class class)co 7135 ↾t crest 16686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-rest 16688 |
This theorem is referenced by: restuni6 41757 restuni5 41758 subsaluni 43000 issmflelem 43378 smfpimltxr 43381 issmfgtlem 43389 issmfgt 43390 issmfgelem 43402 smfpimgtxr 43413 smfresal 43420 |
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