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Mirrors > Home > MPE Home > Th. List > iunopn | Structured version Visualization version GIF version |
Description: The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.) |
Ref | Expression |
---|---|
iunopn | ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiun2g 4966 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | |
2 | 1 | adantl 482 | . 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
3 | uniiunlem 4024 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽)) | |
4 | 3 | ibi 266 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽) |
5 | uniopn 22057 | . . 3 ⊢ ((𝐽 ∈ Top ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽) → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ 𝐽) | |
6 | 4, 5 | sylan2 593 | . 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ 𝐽) |
7 | 2, 6 | eqeltrd 2841 | 1 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 {cab 2717 ∀wral 3066 ∃wrex 3067 ⊆ wss 3892 ∪ cuni 4845 ∪ ciun 4930 Topctop 22053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-in 3899 df-ss 3909 df-pw 4541 df-uni 4846 df-iun 4932 df-top 22054 |
This theorem is referenced by: iincld 22201 tgcn 22414 kgentopon 22700 xkococnlem 22821 qtoptop2 22861 zcld 23987 metnrmlem2 24034 cnambfre 35834 |
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