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Mirrors > Home > MPE Home > Th. List > elutop | Structured version Visualization version GIF version |
Description: Open sets in the topology induced by an uniform structure 𝑈 on 𝑋 (Contributed by Thierry Arnoux, 30-Nov-2017.) |
Ref | Expression |
---|---|
elutop | ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | utopval 23084 | . . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎}) | |
2 | 1 | eleq2d 2816 | . . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ 𝐴 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})) |
3 | sseq2 3913 | . . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ (𝑣 “ {𝑥}) ⊆ 𝐴)) | |
4 | 3 | rexbidv 3206 | . . . . 5 ⊢ (𝑎 = 𝐴 → (∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)) |
5 | 4 | raleqbi1dv 3307 | . . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)) |
6 | 5 | elrab 3591 | . . 3 ⊢ (𝐴 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)) |
7 | 2, 6 | bitrdi 290 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))) |
8 | elex 3416 | . . . . 5 ⊢ (𝐴 ∈ 𝒫 𝑋 → 𝐴 ∈ V) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋 → 𝐴 ∈ V)) |
10 | elfvex 6728 | . . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V) | |
11 | 10 | adantr 484 | . . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ V) |
12 | simpr 488 | . . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
13 | 11, 12 | ssexd 5202 | . . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
14 | 13 | ex 416 | . . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ⊆ 𝑋 → 𝐴 ∈ V)) |
15 | elpwg 4502 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))) |
17 | 9, 14, 16 | pm5.21ndd 384 | . . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
18 | 17 | anbi1d 633 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))) |
19 | 7, 18 | bitrd 282 | 1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 {crab 3055 Vcvv 3398 ⊆ wss 3853 𝒫 cpw 4499 {csn 4527 “ cima 5539 ‘cfv 6358 UnifOncust 23051 unifTopcutop 23082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-iota 6316 df-fun 6360 df-fn 6361 df-fv 6366 df-ust 23052 df-utop 23083 |
This theorem is referenced by: utoptop 23086 utopbas 23087 restutop 23089 restutopopn 23090 ucncn 23136 |
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