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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 36353 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ (topGen‘𝐽) = (topGen‘𝐾))) | 
| 3 | 1 | fneer 36354 | . . . 4 ⊢ ∼ Er V | 
| 4 | 3 | a1i 11 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∼ Er V) | 
| 5 | elex 3501 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V) | |
| 6 | 5 | adantr 480 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → 𝐽 ∈ V) | 
| 7 | 4, 6 | erth 8796 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ [𝐽] ∼ = [𝐾] ∼ )) | 
| 8 | tgtop 22980 | . . 3 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
| 9 | tgtop 22980 | . . 3 ⊢ (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾) | |
| 10 | 8, 9 | eqeqan12d 2751 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ((topGen‘𝐽) = (topGen‘𝐾) ↔ 𝐽 = 𝐾)) | 
| 11 | 2, 7, 10 | 3bitr3d 309 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] ∼ = [𝐾] ∼ ↔ 𝐽 = 𝐾)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 class class class wbr 5143 ◡ccnv 5684 ‘cfv 6561 Er wer 8742 [cec 8743 topGenctg 17482 Topctop 22899 Fnecfne 36337 | 
| 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-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-iun 4993 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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-er 8745 df-ec 8747 df-topgen 17488 df-top 22900 df-fne 36338 | 
| This theorem is referenced by: (None) | 
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