Theorem kgenval 23510

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 23509	. 2 ⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))})
2		unieq 4862	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
3		toponuni 22889	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
4	3	eqcomd 2743	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 = 𝑋)
5	2, 4	sylan9eqr 2794	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
6	5	pweqd 4559	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑗 = 𝐽) → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
7		oveq1 7367	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑘) = (𝐽 ↾t 𝑘))
8	7	eleq1d 2822	. . . . . 6 ⊢ (𝑗 = 𝐽 → ((𝑗 ↾t 𝑘) ∈ Comp ↔ (𝐽 ↾t 𝑘) ∈ Comp))
9	7	eleq2d 2823	. . . . . 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 3312	. . 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 3408	. 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 22888	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15		toponmax 22901	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
16		pwexg 5315	. . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
17		rabexg 5274	. . 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 6950	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 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390  Vcvv 3430   ∩ cin 3889  𝒫 cpw 4542  ∪ cuni 4851  ‘cfv 6492  (class class class)co 7360   ↾_t crest 17374  Topctop 22868  TopOnctopon 22885  Compccmp 23361  𝑘Genckgen 23508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-top 22869  df-topon 22886  df-kgen 23509

This theorem is referenced by:  elkgen  23511  kgentopon  23513