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Mathbox for Jeff Hankins |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > topfneec2 | Structured version Visualization version GIF version |
Description: A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.) |
Ref | Expression |
---|---|
topfneec2.1 | ⊢ ∼ = (Fne ∩ ◡Fne) |
Ref | Expression |
---|---|
topfneec2 | ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] ∼ = [𝐾] ∼ ↔ 𝐽 = 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topfneec2.1 | . . 3 ⊢ ∼ = (Fne ∩ ◡Fne) | |
2 | 1 | fneval 33813 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ (topGen‘𝐽) = (topGen‘𝐾))) |
3 | 1 | fneer 33814 | . . . 4 ⊢ ∼ Er V |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∼ Er V) |
5 | elex 3459 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V) | |
6 | 5 | adantr 484 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → 𝐽 ∈ V) |
7 | 4, 6 | erth 8321 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ [𝐽] ∼ = [𝐾] ∼ )) |
8 | tgtop 21578 | . . 3 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
9 | tgtop 21578 | . . 3 ⊢ (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾) | |
10 | 8, 9 | eqeqan12d 2815 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ((topGen‘𝐽) = (topGen‘𝐾) ↔ 𝐽 = 𝐾)) |
11 | 2, 7, 10 | 3bitr3d 312 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] ∼ = [𝐾] ∼ ↔ 𝐽 = 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 class class class wbr 5030 ◡ccnv 5518 ‘cfv 6324 Er wer 8269 [cec 8270 topGenctg 16703 Topctop 21498 Fnecfne 33797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-er 8272 df-ec 8274 df-topgen 16709 df-top 21499 df-fne 33798 |
This theorem is referenced by: (None) |
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