Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > restuni6 | Structured version Visualization version GIF version |
Description: The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
restuni6.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
restuni6.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
restuni6 | ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restuni6.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | restuni6.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | eqid 2622 | . . . . 5 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
4 | 3 | restin 20970 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↾t 𝐵) = (𝐴 ↾t (𝐵 ∩ ∪ 𝐴))) |
5 | 1, 2, 4 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝐴 ↾t 𝐵) = (𝐴 ↾t (𝐵 ∩ ∪ 𝐴))) |
6 | 5 | unieqd 4446 | . 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = ∪ (𝐴 ↾t (𝐵 ∩ ∪ 𝐴))) |
7 | inss2 3834 | . . . 4 ⊢ (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴 | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴) |
9 | 1, 8 | restuni4 39304 | . 2 ⊢ (𝜑 → ∪ (𝐴 ↾t (𝐵 ∩ ∪ 𝐴)) = (𝐵 ∩ ∪ 𝐴)) |
10 | incom 3805 | . . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵) | |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)) |
12 | 6, 9, 11 | 3eqtrd 2660 | 1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 ∪ cuni 4436 (class class class)co 6650 ↾t crest 16081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-rest 16083 |
This theorem is referenced by: unirestss 39307 |
Copyright terms: Public domain | W3C validator |