Theorem elkgen 21149

Description: Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)

Assertion

Ref	Expression
elkgen	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))

Distinct variable groups:   𝐴,𝑘   𝑘,𝐽   𝑘,𝑋

Proof of Theorem elkgen

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kgenval 21148	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))})
2	1	eleq2d 2673	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ 𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))}))
3		ineq1 3769	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑘) = (𝐴 ∩ 𝑘))
4	3	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘) ↔ (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
5	4	imbi2d 329	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) ↔ ((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))
6	5	ralbidv 2969	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))
7	6	elrab 3331	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))} ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))
8		toponmax 20543	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
9		elpw2g 4754	. . . . 5 ⊢ (𝑋 ∈ 𝐽 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
10	8, 9	syl 17	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
11	10	anbi1d 737	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
12	7, 11	syl5bb 271	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))} ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
13	2, 12	bitrd 267	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ‘cfv 5804  (class class class)co 6549   ↾_t crest 15904  TopOnctopon 20518  Compccmp 20999  𝑘Genckgen 21146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-top 20521  df-topon 20523  df-kgen 21147

This theorem is referenced by:  kgeni  21150  kgentopon  21151  kgenss  21156  kgenidm  21160  iskgen3  21162  kgen2ss  21168  kgencn  21169  kgencn3  21171  txkgen  21265