Theorem kgenval 23451

Description: Value of the compact generator. (The "k" in 𝑘Gen comes from the name "k-space" for these spaces, after the German word kompakt.) (Contributed by Mario Carneiro, 20-Mar-2015.)

Assertion

Ref	Expression
kgenval	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))})

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

Proof of Theorem kgenval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-kgen 23450	. 2 ⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))})
2		unieq 4869	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
3		toponuni 22830	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
4	3	eqcomd 2739	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 = 𝑋)
5	2, 4	sylan9eqr 2790	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
6	5	pweqd 4566	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
7		oveq1 7359	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑘) = (𝐽 ↾t 𝑘))
8	7	eleq1d 2818	. . . . . 6 ⊢ (𝑗 = 𝐽 → ((𝑗 ↾t 𝑘) ∈ Comp ↔ (𝐽 ↾t 𝑘) ∈ Comp))
9	7	eleq2d 2819	. . . . . 6 ⊢ (𝑗 = 𝐽 → ((𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘) ↔ (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
10	8, 9	imbi12d 344	. . . . 5 ⊢ (𝑗 = 𝐽 → (((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)) ↔ ((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))
11	10	adantl 481	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)) ↔ ((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))
12	6, 11	raleqbidv 3313	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → (∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))
13	6, 12	rabeqbidv 3414	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))} = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))})
14		topontop 22829	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15		toponmax 22842	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
16		pwexg 5318	. . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
17		rabexg 5277	. . 3 ⊢ (𝒫 𝑋 ∈ V → {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))} ∈ V)
18	15, 16, 17	3syl 18	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))} ∈ V)
19	1, 13, 14, 18	fvmptd2 6943	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  {crab 3396  Vcvv 3437   ∩ cin 3897  𝒫 cpw 4549  ∪ cuni 4858  ‘cfv 6486  (class class class)co 7352   ↾_t crest 17326  Topctop 22809  TopOnctopon 22826  Compccmp 23302  𝑘Genckgen 23449

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-top 22810  df-topon 22827  df-kgen 23450

This theorem is referenced by:  elkgen  23452  kgentopon  23454