Theorem topfneec2 37271

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 37267	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ (topGen‘𝐽) = (topGen‘𝐾)))
3	1	fneer 37268	. . . 4 ⊢ ∼ Er V
4	3	a1i 11	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∼ Er V)
5		elex 3474	. . . 4 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V)
6	5	adantr 486	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → 𝐽 ∈ V)
7	4, 6	erth 8783	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∼ 𝐾 ↔ [𝐽] ∼ = [𝐾] ∼ ))
8		tgtop 23303	. . 3 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
9		tgtop 23303	. . 3 ⊢ (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾)
10	8, 9	eqeqan12d 2777	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  Vcvv 3453   ∩ cin 4091   class class class wbr 5105  ^◡ccnv 5653  ‘cfv 6546   Er wer 8725  [cec 8726  topGenctg 17620  Topctop 23223  Fnecfne 37251

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fv 6554  df-er 8728  df-ec 8730  df-topgen 17626  df-top 23224  df-fne 37252

This theorem is used by: (None)