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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 4148 | . . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)) |
3 | restuni4.2 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴) | |
4 | dfss 3916 | . . 3 ⊢ (𝐵 ⊆ ∪ 𝐴 ↔ 𝐵 = (𝐵 ∩ ∪ 𝐴)) | |
5 | 3, 4 | sylib 217 | . 2 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ ∪ 𝐴)) |
6 | restuni4.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | 6 | uniexd 7657 | . . . 4 ⊢ (𝜑 → ∪ 𝐴 ∈ V) |
8 | 7, 3 | ssexd 5268 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
9 | 6, 8 | restuni3 42996 | . 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) |
10 | 2, 5, 9 | 3eqtr4rd 2787 | 1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ∩ cin 3897 ⊆ wss 3898 ∪ cuni 4852 (class class class)co 7337 ↾t crest 17228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-rest 17230 |
This theorem is referenced by: restuni6 43000 restuni5 43001 subsaluni 44243 issmflelem 44627 issmfgtlem 44638 issmfgt 44639 issmfgelem 44652 smfresal 44671 |
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