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| Mirrors > Home > MPE Home > Th. List > Mathboxes > opncldeqv | Structured version Visualization version GIF version | ||
| Description: Conditions on open sets are equivalent to conditions on closed sets. (Contributed by Zhi Wang, 30-Aug-2024.) | 
| Ref | Expression | 
|---|---|
| opncldeqv.1 | ⊢ (𝜑 → 𝐽 ∈ Top) | 
| opncldeqv.2 | ⊢ ((𝜑 ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| opncldeqv | ⊢ (𝜑 → (∀𝑥 ∈ 𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 2 | 1 | cldopn 23039 | . . 3 ⊢ (𝑦 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ 𝐽) | 
| 3 | 2 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑦) ∈ 𝐽) | 
| 4 | opncldeqv.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) | |
| 5 | 1 | opncld 23041 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)) | 
| 6 | elssuni 4937 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽) | |
| 7 | dfss4 4269 | . . . . . . . . 9 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥) | |
| 8 | 6, 7 | sylib 218 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥) | 
| 9 | 8 | eqcomd 2743 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐽 → 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))) | 
| 10 | 9 | adantl 481 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))) | 
| 11 | 5, 10 | jca 511 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))) | 
| 12 | eleq1 2829 | . . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑦 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))) | |
| 13 | difeq2 4120 | . . . . . . 7 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (∪ 𝐽 ∖ 𝑦) = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))) | |
| 14 | 13 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑥 = (∪ 𝐽 ∖ 𝑦) ↔ 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))) | 
| 15 | 12, 14 | anbi12d 632 | . . . . 5 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → ((𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) ↔ ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))))) | 
| 16 | 5, 11, 15 | spcedv 3598 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦))) | 
| 17 | df-rex 3071 | . . . 4 ⊢ (∃𝑦 ∈ (Clsd‘𝐽)𝑥 = (∪ 𝐽 ∖ 𝑦) ↔ ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦))) | |
| 18 | 16, 17 | sylibr 234 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → ∃𝑦 ∈ (Clsd‘𝐽)𝑥 = (∪ 𝐽 ∖ 𝑦)) | 
| 19 | 4, 18 | sylan 580 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ∃𝑦 ∈ (Clsd‘𝐽)𝑥 = (∪ 𝐽 ∖ 𝑦)) | 
| 20 | opncldeqv.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) → (𝜓 ↔ 𝜒)) | |
| 21 | 3, 19, 20 | ralxfrd 5408 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∖ cdif 3948 ⊆ wss 3951 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 Clsdccld 23024 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-top 22900 df-cld 23027 | 
| This theorem is referenced by: (None) | 
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