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Theorem isr0 22347

Description: The property "𝐽 is an R₀ space". A space is R₀ if any two topologically distinguishable points are separated (there is an open set containing each one and disjoint from the other). Or in contraposition, if every open set which contains 𝑥 also contains 𝑦, so there is no separation, then 𝑥 and 𝑦 are members of the same open sets. We have chosen not to give this definition a name, because it turns out that a space is R₀ if and only if its Kolmogorov quotient is T₁, so that is what we prove here. (Contributed by Mario Carneiro, 25-Aug-2015.)

**Hypothesis**
Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

**Assertion**
Ref	Expression
isr0	⊢ (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜))))

Distinct variable groups: 𝑤,𝑜,𝑥,𝑦,𝑧,𝐽 𝑜,𝐹,𝑤,𝑧 𝑜,𝑋,𝑤,𝑥,𝑦,𝑧 Allowed substitution hints: 𝐹(𝑥,𝑦)

Proof of Theorem isr0 Dummy variables 𝑎 𝑏 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqid 22338	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
3	2	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
4		cnima 21875	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑣) ∈ 𝐽)
5	3, 4	sylan 582	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑣) ∈ 𝐽)
6		eleq2 2903	. . . . . . . . . . 11 ⊢ (𝑜 = (^◡𝐹 “ 𝑣) → (𝑧 ∈ 𝑜 ↔ 𝑧 ∈ (^◡𝐹 “ 𝑣)))
7		eleq2 2903	. . . . . . . . . . 11 ⊢ (𝑜 = (^◡𝐹 “ 𝑣) → (𝑤 ∈ 𝑜 ↔ 𝑤 ∈ (^◡𝐹 “ 𝑣)))
8	6, 7	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑜 = (^◡𝐹 “ 𝑣) → ((𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) ↔ (𝑧 ∈ (^◡𝐹 “ 𝑣) → 𝑤 ∈ (^◡𝐹 “ 𝑣))))
9	8	rspcv 3620	. . . . . . . . 9 ⊢ ((^◡𝐹 “ 𝑣) ∈ 𝐽 → (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → (𝑧 ∈ (^◡𝐹 “ 𝑣) → 𝑤 ∈ (^◡𝐹 “ 𝑣))))
10	5, 9	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → (𝑧 ∈ (^◡𝐹 “ 𝑣) → 𝑤 ∈ (^◡𝐹 “ 𝑣))))
11	1	kqffn 22335	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
12	11	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐹 Fn 𝑋)
13	12	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋)
14		fnfun 6455	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → Fun 𝐹)
16		simprl 769	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
17	16	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧 ∈ 𝑋)
18		fndm 6457	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
19	13, 18	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → dom 𝐹 = 𝑋)
20	17, 19	eleqtrrd 2918	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧 ∈ dom 𝐹)
21		fvimacnv 6825	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → ((𝐹‘𝑧) ∈ 𝑣 ↔ 𝑧 ∈ (^◡𝐹 “ 𝑣)))
22	15, 20, 21	syl2anc 586	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹‘𝑧) ∈ 𝑣 ↔ 𝑧 ∈ (^◡𝐹 “ 𝑣)))
23		simprr 771	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑤 ∈ 𝑋)
24	23	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤 ∈ 𝑋)
25	24, 19	eleqtrrd 2918	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤 ∈ dom 𝐹)
26		fvimacnv 6825	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑤 ∈ dom 𝐹) → ((𝐹‘𝑤) ∈ 𝑣 ↔ 𝑤 ∈ (^◡𝐹 “ 𝑣)))
27	15, 25, 26	syl2anc 586	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹‘𝑤) ∈ 𝑣 ↔ 𝑤 ∈ (^◡𝐹 “ 𝑣)))
28	22, 27	imbi12d 347	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) ↔ (𝑧 ∈ (^◡𝐹 “ 𝑣) → 𝑤 ∈ (^◡𝐹 “ 𝑣))))
29	10, 28	sylibrd 261	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣)))
30	29	ralrimdva 3191	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣)))
31		simplr 767	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (KQ‘𝐽) ∈ Fre)
32		fnfvelrn 6850	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran 𝐹)
33	12, 16, 32	syl2anc 586	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) ∈ ran 𝐹)
34	1	kqtopon 22337	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
35	34	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
36		toponuni 21524	. . . . . . . . 9 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = ∪ (KQ‘𝐽))
37	35, 36	syl 17	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ran 𝐹 = ∪ (KQ‘𝐽))
38	33, 37	eleqtrd 2917	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) ∈ ∪ (KQ‘𝐽))
39		fnfvelrn 6850	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑤) ∈ ran 𝐹)
40	12, 23, 39	syl2anc 586	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) ∈ ran 𝐹)
41	40, 37	eleqtrd 2917	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) ∈ ∪ (KQ‘𝐽))
42		eqid 2823	. . . . . . . 8 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
43	42	t1sep2 21979	. . . . . . 7 ⊢ (((KQ‘𝐽) ∈ Fre ∧ (𝐹‘𝑧) ∈ ∪ (KQ‘𝐽) ∧ (𝐹‘𝑤) ∈ ∪ (KQ‘𝐽)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝐹‘𝑧) = (𝐹‘𝑤)))
44	31, 38, 41, 43	syl3anc 1367	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝐹‘𝑧) = (𝐹‘𝑤)))
45	30, 44	syld 47	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → (𝐹‘𝑧) = (𝐹‘𝑤)))
46	1	kqfeq 22334	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
47		eleq2 2903	. . . . . . . . . 10 ⊢ (𝑜 = 𝑦 → (𝑧 ∈ 𝑜 ↔ 𝑧 ∈ 𝑦))
48		eleq2 2903	. . . . . . . . . 10 ⊢ (𝑜 = 𝑦 → (𝑤 ∈ 𝑜 ↔ 𝑤 ∈ 𝑦))
49	47, 48	bibi12d 348	. . . . . . . . 9 ⊢ (𝑜 = 𝑦 → ((𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜) ↔ (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
50	49	cbvralvw 3451	. . . . . . . 8 ⊢ (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦))
51	46, 50	syl6bbr 291	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
52	51	3expb 1116	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
53	52	adantlr 713	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
54	45, 53	sylibd 241	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
55	54	ralrimivva 3193	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
56	55	ex 415	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜))))
57	1	kqopn 22344	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜 ∈ 𝐽) → (𝐹 “ 𝑜) ∈ (KQ‘𝐽))
58	57	ad4ant14 750	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝐹 “ 𝑜) ∈ (KQ‘𝐽))
59		eleq2 2903	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝐹 “ 𝑜) → ((𝐹‘𝑧) ∈ 𝑣 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑜)))
60		eleq2 2903	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝐹 “ 𝑜) → ((𝐹‘𝑤) ∈ 𝑣 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑜)))
61	59, 60	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑣 = (𝐹 “ 𝑜) → (((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑜) → (𝐹‘𝑤) ∈ (𝐹 “ 𝑜))))
62	61	rspcv 3620	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑜) ∈ (KQ‘𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑜) → (𝐹‘𝑤) ∈ (𝐹 “ 𝑜))))
63	58, 62	syl 17	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑜) → (𝐹‘𝑤) ∈ (𝐹 “ 𝑜))))
64	1	kqfvima 22340	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜 ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑜 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑜)))
65	64	3expa 1114	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑜 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑜)))
66	65	an32s 650	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑧 ∈ 𝑜 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑜)))
67	66	adantlr 713	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑧 ∈ 𝑜 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑜)))
68	1	kqfvima 22340	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜 ∈ 𝐽 ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝑜 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑜)))
69	68	3expa 1114	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝑜 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑜)))
70	69	an32s 650	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑤 ∈ 𝑜 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑜)))
71	70	adantllr 717	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑤 ∈ 𝑜 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑜)))
72	67, 71	imbi12d 347	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → ((𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑜) → (𝐹‘𝑤) ∈ (𝐹 “ 𝑜))))
73	63, 72	sylibrd 261	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜)))
74	73	ralrimdva 3191	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜)))
75	1	kqfval 22333	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) = {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦})
76	75	adantr 483	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑧) = {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦})
77	1	kqfval 22333	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑤) = {𝑦 ∈ 𝐽 ∣ 𝑤 ∈ 𝑦})
78	77	adantlr 713	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑤) = {𝑦 ∈ 𝐽 ∣ 𝑤 ∈ 𝑦})
79	76, 78	eqeq12d 2839	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝑤 ∈ 𝑦}))
80		rabbi 3385	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) ↔ {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝑤 ∈ 𝑦})
81	50, 80	bitri 277	. . . . . . . . 9 ⊢ (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜) ↔ {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝑤 ∈ 𝑦})
82	79, 81	syl6bbr 291	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
83	82	biimprd 250	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜) → (𝐹‘𝑧) = (𝐹‘𝑤)))
84	74, 83	imim12d 81	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → ((∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝐹‘𝑧) = (𝐹‘𝑤))))
85	84	ralimdva 3179	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) → (∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)) → ∀𝑤 ∈ 𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝐹‘𝑧) = (𝐹‘𝑤))))
86	85	ralimdva 3179	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)) → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝐹‘𝑧) = (𝐹‘𝑤))))
87		eleq1 2902	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 ∈ 𝑣 ↔ (𝐹‘𝑧) ∈ 𝑣))
88	87	imbi1d 344	. . . . . . . . . 10 ⊢ (𝑎 = (𝐹‘𝑧) → ((𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) ↔ ((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣)))
89	88	ralbidv 3199	. . . . . . . . 9 ⊢ (𝑎 = (𝐹‘𝑧) → (∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣)))
90		eqeq1 2827	. . . . . . . . 9 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 = 𝑏 ↔ (𝐹‘𝑧) = 𝑏))
91	89, 90	imbi12d 347	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑧) → ((∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) → 𝑎 = 𝑏) ↔ (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣) → (𝐹‘𝑧) = 𝑏)))
92	91	ralbidv 3199	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑧) → (∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) → 𝑎 = 𝑏) ↔ ∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣) → (𝐹‘𝑧) = 𝑏)))
93	92	ralrn 6856	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) → 𝑎 = 𝑏) ↔ ∀𝑧 ∈ 𝑋 ∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣) → (𝐹‘𝑧) = 𝑏)))
94		eleq1 2902	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐹‘𝑤) → (𝑏 ∈ 𝑣 ↔ (𝐹‘𝑤) ∈ 𝑣))
95	94	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑤) → (((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣) ↔ ((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣)))
96	95	ralbidv 3199	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑤) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣)))
97		eqeq2 2835	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑤) → ((𝐹‘𝑧) = 𝑏 ↔ (𝐹‘𝑧) = (𝐹‘𝑤)))
98	96, 97	imbi12d 347	. . . . . . . 8 ⊢ (𝑏 = (𝐹‘𝑤) → ((∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣) → (𝐹‘𝑧) = 𝑏) ↔ (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝐹‘𝑧) = (𝐹‘𝑤))))
99	98	ralrn 6856	. . . . . . 7 ⊢ (𝐹 Fn 𝑋 → (∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣) → (𝐹‘𝑧) = 𝑏) ↔ ∀𝑤 ∈ 𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝐹‘𝑧) = (𝐹‘𝑤))))
100	99	ralbidv 3199	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (∀𝑧 ∈ 𝑋 ∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣) → (𝐹‘𝑧) = 𝑏) ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝐹‘𝑧) = (𝐹‘𝑤))))
101	93, 100	bitrd 281	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) → 𝑎 = 𝑏) ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝐹‘𝑧) = (𝐹‘𝑤))))
102	11, 101	syl 17	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) → 𝑎 = 𝑏) ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝐹‘𝑧) = (𝐹‘𝑤))))
103	86, 102	sylibrd 261	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)) → ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) → 𝑎 = 𝑏)))
104		ist1-2 21957	. . . 4 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) → 𝑎 = 𝑏)))
105	34, 104	syl 17	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) → 𝑎 = 𝑏)))
106	103, 105	sylibrd 261	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)) → (KQ‘𝐽) ∈ Fre))
107	56, 106	impbid 214	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 ∪ cuni 4840 ↦ cmpt 5148 ^◡ccnv 5556 dom cdm 5557 ran crn 5558 “ cima 5560 Fun wfun 6351 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 TopOnctopon 21520 Cn ccn 21834 Frect1 21917 KQckq 22303

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-rep 5192 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-reu 3147 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-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-topgen 16719 df-qtop 16782 df-top 21504 df-topon 21521 df-cld 21629 df-cn 21837 df-t1 21924 df-kq 22304

This theorem is referenced by: r0sep 22358

W3C validator