Theorem topfneec 32350

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 32348	. . . 4 ⊢ ∼ Er V
3		errel 7751	. . . 4 ⊢ ( ∼ Er V → Rel ∼ )
4	2, 3	ax-mp 5	. . 3 ⊢ Rel ∼
5		relelec 7787	. . 3 ⊢ (Rel ∼ → (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴))
6	4, 5	ax-mp 5	. 2 ⊢ (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴)
7	4	brrelex2i 5159	. . . 4 ⊢ (𝐽 ∼ 𝐴 → 𝐴 ∈ V)
8	7	a1i 11	. . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 → 𝐴 ∈ V))
9		eleq1 2689	. . . . . . 7 ⊢ ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top))
10	9	biimparc 504	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top)
11		tgclb 20774	. . . . . 6 ⊢ (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top)
12	10, 11	sylibr 224	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases)
13		elex 3212	. . . . 5 ⊢ (𝐴 ∈ TopBases → 𝐴 ∈ V)
14	12, 13	syl 17	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ V)
15	14	ex 450	. . 3 ⊢ (𝐽 ∈ Top → ((topGen‘𝐴) = 𝐽 → 𝐴 ∈ V))
16	1	fneval 32347	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴)))
17		tgtop 20777	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
18	17	eqeq1d 2624	. . . . . . 7 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴)))
19		eqcom 2629	. . . . . . 7 ⊢ (𝐽 = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)
20	18, 19	syl6bb 276	. . . . . 6 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
21	20	adantr 481	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
22	16, 21	bitrd 268	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽))
23	22	ex 450	. . 3 ⊢ (𝐽 ∈ Top → (𝐴 ∈ V → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽)))
24	8, 15, 23	pm5.21ndd 369	. 2 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽))
25	6, 24	syl5bb 272	1 ⊢ (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ∼ ↔ (topGen‘𝐴) = 𝐽))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∩ cin 3573   class class class wbr 4653  ^◡ccnv 5113  Rel wrel 5119  ‘cfv 5888   Er wer 7739  [cec 7740  topGenctg 16098  Topctop 20698  TopBasesctb 20749  Fnecfne 32331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-er 7742  df-ec 7744  df-topgen 16104  df-top 20699  df-bases 20750  df-fne 32332

This theorem is referenced by: (None)