Theorem r0sep 23642

Description: The separation property of an R₀ space. (Contributed by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
r0sep	⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜)))

Distinct variable groups:   𝐴,𝑜   𝐵,𝑜   𝑜,𝐽   𝑜,𝑋

Proof of Theorem r0sep

Dummy variables 𝑥 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. . . 4 ⊢ (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤}) = (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤})
2	1	isr0 23631	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜))))
3	2	biimpa 476	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)))
4		eleq1 2817	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜))
5	4	imbi1d 341	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜)))
6	5	ralbidv 3157	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜)))
7	4	bibi1d 343	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)))
8	7	ralbidv 3157	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)))
9	6, 8	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)) ↔ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜))))
10		eleq1 2817	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))
11	10	imbi2d 340	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜)))
12	11	ralbidv 3157	. . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜)))
13	10	bibi2d 342	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜)))
14	13	ralbidv 3157	. . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜)))
15	12, 14	imbi12d 344	. . 3 ⊢ (𝑦 = 𝐵 → ((∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)) ↔ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))))
16	9, 15	rspc2v 3602	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))))
17	3, 16	mpan9 506	1 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408   ↦ cmpt 5191  ‘cfv 6514  TopOnctopon 22804  Frect1 23201  KQckq 23587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-topgen 17413  df-qtop 17477  df-top 22788  df-topon 22805  df-cld 22913  df-cn 23121  df-t1 23208  df-kq 23588

This theorem is referenced by: (None)