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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 4034 | . . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)) |
3 | restuni4.2 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴) | |
4 | dfss 3813 | . . 3 ⊢ (𝐵 ⊆ ∪ 𝐴 ↔ 𝐵 = (𝐵 ∩ ∪ 𝐴)) | |
5 | 3, 4 | sylib 210 | . 2 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ ∪ 𝐴)) |
6 | restuni4.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | 6 | uniexd 40097 | . . . 4 ⊢ (𝜑 → ∪ 𝐴 ∈ V) |
8 | 7, 3 | ssexd 5032 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
9 | 6, 8 | restuni3 40115 | . 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) |
10 | 2, 5, 9 | 3eqtr4rd 2872 | 1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∩ cin 3797 ⊆ wss 3798 ∪ cuni 4660 (class class class)co 6910 ↾t crest 16441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-rest 16443 |
This theorem is referenced by: restuni6 40119 restuni5 40120 subsaluni 41367 issmflelem 41745 smfpimltxr 41748 issmfgtlem 41756 issmfgt 41757 issmfgelem 41769 smfpimgtxr 41780 smfresal 41787 |
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