topfneec2 - Mathbox for Jeff Hankins

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Theorem topfneec2 34545

Description: A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.)

**Hypothesis**
Ref	Expression
topfneec2.1	⊢ ∼ = (Fne ∩ ^◡Fne)

**Assertion**
Ref	Expression
topfneec2	⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] ∼ = [𝐾] ∼ ↔ 𝐽 = 𝐾))

**Proof of Theorem topfneec2**
Step	Hyp	Ref	Expression
1		topfneec2.1	. . 3 ⊢ ∼ = (Fne ∩ ^◡Fne)
2	1	fneval 34541	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ (topGen‘𝐽) = (topGen‘𝐾)))
3	1	fneer 34542	. . . 4 ⊢ ∼ Er V
4	3	a1i 11	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∼ Er V)
5		elex 3450	. . . 4 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V)
6	5	adantr 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → 𝐽 ∈ V)
7	4, 6	erth 8547	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ [𝐽] ∼ = [𝐾] ∼ ))
8		tgtop 22123	. . 3 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
9		tgtop 22123	. . 3 ⊢ (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾)
10	8, 9	eqeqan12d 2752	. 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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∩ cin 3886 class class class wbr 5074 ^◡ccnv 5588 ‘cfv 6433 Er wer 8495 [cec 8496 topGenctg 17148 Topctop 22042 Fnecfne 34525

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fv 6441 df-er 8498 df-ec 8500 df-topgen 17154 df-top 22043 df-fne 34526

This theorem is referenced by: (None)

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