Theorem topfneec 33705

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 33703	. . . 4 ⊢ ∼ Er V
3		errel 8300	. . . 4 ⊢ ( ∼ Er V → Rel ∼ )
4	2, 3	ax-mp 5	. . 3 ⊢ Rel ∼
5		relelec 8336	. . 3 ⊢ (Rel ∼ → (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴))
6	4, 5	ax-mp 5	. 2 ⊢ (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴)
7	4	brrelex2i 5611	. . . 4 ⊢ (𝐽 ∼ 𝐴 → 𝐴 ∈ V)
8	7	a1i 11	. . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 → 𝐴 ∈ V))
9		eleq1 2902	. . . . . . 7 ⊢ ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top))
10	9	biimparc 482	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top)
11		tgclb 21580	. . . . . 6 ⊢ (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top)
12	10, 11	sylibr 236	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases)
13		elex 3514	. . . . 5 ⊢ (𝐴 ∈ TopBases → 𝐴 ∈ V)
14	12, 13	syl 17	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ V)
15	14	ex 415	. . 3 ⊢ (𝐽 ∈ Top → ((topGen‘𝐴) = 𝐽 → 𝐴 ∈ V))
16	1	fneval 33702	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴)))
17		tgtop 21583	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
18	17	eqeq1d 2825	. . . . . . 7 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴)))
19		eqcom 2830	. . . . . . 7 ⊢ (𝐽 = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)
20	18, 19	syl6bb 289	. . . . . 6 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
21	20	adantr 483	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
22	16, 21	bitrd 281	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽))
23	22	ex 415	. . 3 ⊢ (𝐽 ∈ Top → (𝐴 ∈ V → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽)))
24	8, 15, 23	pm5.21ndd 383	. 2 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽))
25	6, 24	syl5bb 285	1 ⊢ (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ∼ ↔ (topGen‘𝐴) = 𝐽))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ∩ cin 3937   class class class wbr 5068  ^◡ccnv 5556  Rel wrel 5562  ‘cfv 6357   Er wer 8288  [cec 8289  topGenctg 16713  Topctop 21503  TopBasesctb 21555  Fnecfne 33686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fv 6365  df-er 8291  df-ec 8293  df-topgen 16719  df-top 21504  df-bases 21556  df-fne 33687

This theorem is referenced by: (None)