Theorem topfneec2 32947

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 32943	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ (topGen‘𝐽) = (topGen‘𝐾)))
3	1	fneer 32944	. . . 4 ⊢ ∼ Er V
4	3	a1i 11	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∼ Er V)
5		elex 3414	. . . 4 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V)
6	5	adantr 474	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → 𝐽 ∈ V)
7	4, 6	erth 8075	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ [𝐽] ∼ = [𝐾] ∼ ))
8		tgtop 21196	. . 3 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
9		tgtop 21196	. . 3 ⊢ (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾)
10	8, 9	eqeqan12d 2794	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ((topGen‘𝐽) = (topGen‘𝐾) ↔ 𝐽 = 𝐾))
11	2, 7, 10	3bitr3d 301	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] ∼ = [𝐾] ∼ ↔ 𝐽 = 𝐾))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ∩ cin 3791   class class class wbr 4888  ^◡ccnv 5356  ‘cfv 6137   Er wer 8025  [cec 8026  topGenctg 16495  Topctop 21116  Fnecfne 32927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fv 6145  df-er 8028  df-ec 8030  df-topgen 16501  df-top 21117  df-fne 32928

This theorem is referenced by: (None)