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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 33702 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ (topGen‘𝐽) = (topGen‘𝐾))) |
3 | 1 | fneer 33703 | . . . 4 ⊢ ∼ Er V |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∼ Er V) |
5 | elex 3514 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V) | |
6 | 5 | adantr 483 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → 𝐽 ∈ V) |
7 | 4, 6 | erth 8340 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ [𝐽] ∼ = [𝐾] ∼ )) |
8 | tgtop 21583 | . . 3 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
9 | tgtop 21583 | . . 3 ⊢ (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾) | |
10 | 8, 9 | eqeqan12d 2840 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ((topGen‘𝐽) = (topGen‘𝐾) ↔ 𝐽 = 𝐾)) |
11 | 2, 7, 10 | 3bitr3d 311 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] ∼ = [𝐾] ∼ ↔ 𝐽 = 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∩ cin 3937 class class class wbr 5068 ◡ccnv 5556 ‘cfv 6357 Er wer 8288 [cec 8289 topGenctg 16713 Topctop 21503 Fnecfne 33686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 df-er 8291 df-ec 8293 df-topgen 16719 df-top 21504 df-fne 33687 |
This theorem is referenced by: (None) |
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