Theorem topfneec 36977

Description: A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Hypothesis

Ref	Expression
topfneec.1	⊢ ∼ = (Fne ∩ ◡Fne)

Assertion

Ref	Expression
topfneec	⊢ (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ∼ ↔ (topGen‘𝐴) = 𝐽))

Proof of Theorem topfneec

Step	Hyp	Ref	Expression
1		topfneec.1	. . . . 5 ⊢ ∼ = (Fne ∩ ◡Fne)
2	1	fneer 36975	. . . 4 ⊢ ∼ Er V
3		errel 8709	. . . 4 ⊢ ( ∼ Er V → Rel ∼ )
4	2, 3	ax-mp 5	. . 3 ⊢ Rel ∼
5		relelec 8747	. . 3 ⊢ (Rel ∼ → (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴))
6	4, 5	ax-mp 5	. 2 ⊢ (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴)
7	4	brrelex2i 5712	. . . 4 ⊢ (𝐽 ∼ 𝐴 → 𝐴 ∈ V)
8	7	a1i 11	. . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 → 𝐴 ∈ V))
9		eleq1 2848	. . . . . . 7 ⊢ ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top))
10	9	biimparc 485	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top)
11		tgclb 23198	. . . . . 6 ⊢ (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top)
12	10, 11	sylibr 237	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases)
13		elex 3471	. . . . 5 ⊢ (𝐴 ∈ TopBases → 𝐴 ∈ V)
14	12, 13	syl 18	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ V)
15	14	ex 418	. . 3 ⊢ (𝐽 ∈ Top → ((topGen‘𝐴) = 𝐽 → 𝐴 ∈ V))
16	1	fneval 36974	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴)))
17		tgtop 23201	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
18	17	eqeq1d 2762	. . . . . . 7 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴)))
19		eqcom 2767	. . . . . . 7 ⊢ (𝐽 = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)
20	18, 19	bitrdi 290	. . . . . 6 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
21	20	adantr 486	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
22	16, 21	bitrd 282	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽))
23	22	ex 418	. . 3 ⊢ (𝐽 ∈ Top → (𝐴 ∈ V → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽)))
24	8, 15, 23	pm5.21ndd 382	. 2 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽))
25	6, 24	bitrid 286	1 ⊢ (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ∼ ↔ (topGen‘𝐴) = 𝐽))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∩ cin 3898   class class class wbr 5103  ^◡ccnv 5654  Rel wrel 5660  ‘cfv 6533   Er wer 8696  [cec 8697  topGenctg 17525  Topctop 23121  TopBasesctb 23173  Fnecfne 36958

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541  df-er 8699  df-ec 8701  df-topgen 17531  df-top 23122  df-bases 23174  df-fne 36959

This theorem is used by: (None)