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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 3838 | . . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)) |
3 | restuni4.2 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴) | |
4 | dfss 3622 | . . 3 ⊢ (𝐵 ⊆ ∪ 𝐴 ↔ 𝐵 = (𝐵 ∩ ∪ 𝐴)) | |
5 | 3, 4 | sylib 208 | . 2 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ ∪ 𝐴)) |
6 | restuni4.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | 6 | uniexd 39595 | . . . 4 ⊢ (𝜑 → ∪ 𝐴 ∈ V) |
8 | 7, 3 | ssexd 4838 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
9 | 6, 8 | restuni3 39615 | . 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) |
10 | 2, 5, 9 | 3eqtr4rd 2696 | 1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∩ cin 3606 ⊆ wss 3607 ∪ cuni 4468 (class class class)co 6690 ↾t crest 16128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-rest 16130 |
This theorem is referenced by: restuni6 39619 restuni5 39620 subsaluni 40896 issmflelem 41274 smfpimltxr 41277 issmfgtlem 41285 issmfgt 41286 issmfgelem 41298 smfpimgtxr 41309 smfresal 41316 |
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