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Theorem isr0 24111

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 24102	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
3	2	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
4		cnima 23638	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑣) ∈ 𝐽)
5	3, 4	sylan 592	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑣) ∈ 𝐽)
6		eleq2 2853	. . . . . . . . . . 11 ⊢ (𝑜 = (^◡𝐹 “ 𝑣) → (𝑧 ∈ 𝑜 ↔ 𝑧 ∈ (^◡𝐹 “ 𝑣)))
7		eleq2 2853	. . . . . . . . . . 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 3576	. . . . . . . . 9 ⊢ ((^◡𝐹 “ 𝑣) ∈ 𝐽 → (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → (𝑧 ∈ (^◡𝐹 “ 𝑣) → 𝑤 ∈ (^◡𝐹 “ 𝑣))))
10	5, 9	syl 18	. . . . . . . 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 24099	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
12	11	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐹 Fn 𝑋)
13	12	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋)
14		fnfun 6669	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
15	13, 14	syl 18	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → Fun 𝐹)
16		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
17	16	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧 ∈ 𝑋)
18	13	fndmd 6674	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → dom 𝐹 = 𝑋)
19	17, 18	eleqtrrd 2867	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧 ∈ dom 𝐹)
20		fvimacnv 7083	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → ((𝐹‘𝑧) ∈ 𝑣 ↔ 𝑧 ∈ (^◡𝐹 “ 𝑣)))
21	15, 19, 20	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹‘𝑧) ∈ 𝑣 ↔ 𝑧 ∈ (^◡𝐹 “ 𝑣)))
22		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑤 ∈ 𝑋)
23	22	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤 ∈ 𝑋)
24	23, 18	eleqtrrd 2867	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤 ∈ dom 𝐹)
25		fvimacnv 7083	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑤 ∈ dom 𝐹) → ((𝐹‘𝑤) ∈ 𝑣 ↔ 𝑤 ∈ (^◡𝐹 “ 𝑣)))
26	15, 24, 25	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹‘𝑤) ∈ 𝑣 ↔ 𝑤 ∈ (^◡𝐹 “ 𝑣)))
27	21, 26	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))))
28	10, 27	sylibrd 262	. . . . . . 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)))
29	28	ralrimdva 3166	. . . . . 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)))
30		simplr 781	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (KQ‘𝐽) ∈ Fre)
31		fnfvelrn 7112	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ran 𝐹)
32	12, 16, 31	syl2anc 596	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) ∈ ran 𝐹)
33	1	kqtopon 24101	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
34	33	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
35		toponuni 23287	. . . . . . . . 9 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = ∪ (KQ‘𝐽))
36	34, 35	syl 18	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ran 𝐹 = ∪ (KQ‘𝐽))
37	32, 36	eleqtrd 2866	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) ∈ ∪ (KQ‘𝐽))
38		fnfvelrn 7112	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑤) ∈ ran 𝐹)
39	12, 22, 38	syl2anc 596	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) ∈ ran 𝐹)
40	39, 36	eleqtrd 2866	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) ∈ ∪ (KQ‘𝐽))
41		eqid 2763	. . . . . . . 8 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
42	41	t1sep2 23742	. . . . . . 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)))
43	30, 37, 40, 42	syl3anc 1399	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝐹‘𝑧) = (𝐹‘𝑤)))
44	29, 43	syld 48	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → (𝐹‘𝑧) = (𝐹‘𝑤)))
45	1	kqfeq 24098	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
46		eleq2 2853	. . . . . . . . . 10 ⊢ (𝑜 = 𝑦 → (𝑧 ∈ 𝑜 ↔ 𝑧 ∈ 𝑦))
47		eleq2 2853	. . . . . . . . . 10 ⊢ (𝑜 = 𝑦 → (𝑤 ∈ 𝑜 ↔ 𝑤 ∈ 𝑦))
48	46, 47	bibi12d 348	. . . . . . . . 9 ⊢ (𝑜 = 𝑦 → ((𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜) ↔ (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
49	48	cbvralvw 3244	. . . . . . . 8 ⊢ (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦))
50	45, 49	bitr4di 292	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
51	50	3expb 1138	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
52	51	adantlr 728	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
53	44, 52	sylibd 242	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
54	53	ralrimivva 3209	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
55	54	ex 418	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜))))
56	1	kqopn 24108	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜 ∈ 𝐽) → (𝐹 “ 𝑜) ∈ (KQ‘𝐽))
57	56	ad4ant14 765	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝐹 “ 𝑜) ∈ (KQ‘𝐽))
58		eleq2 2853	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝐹 “ 𝑜) → ((𝐹‘𝑧) ∈ 𝑣 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑜)))
59		eleq2 2853	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝐹 “ 𝑜) → ((𝐹‘𝑤) ∈ 𝑣 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑜)))
60	58, 59	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑣 = (𝐹 “ 𝑜) → (((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑜) → (𝐹‘𝑤) ∈ (𝐹 “ 𝑜))))
61	60	rspcv 3576	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑜) ∈ (KQ‘𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → ((𝐹‘𝑧) ∈ (𝐹 “ 𝑜) → (𝐹‘𝑤) ∈ (𝐹 “ 𝑜))))
62	57, 61	syl 18	. . . . . . . . 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))))
63	1	kqfvima 24104	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜 ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑜 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑜)))
64	63	3expa 1136	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑜 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑜)))
65	64	an32s 665	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑧 ∈ 𝑜 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑜)))
66	65	adantlr 728	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑧 ∈ 𝑜 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑜)))
67	1	kqfvima 24104	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜 ∈ 𝐽 ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝑜 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑜)))
68	67	3expa 1136	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝑜 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑜)))
69	68	an32s 665	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑤 ∈ 𝑜 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑜)))
70	69	adantllr 732	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑤 ∈ 𝑜 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑜)))
71	66, 70	imbi12d 347	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → ((𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑜) → (𝐹‘𝑤) ∈ (𝐹 “ 𝑜))))
72	62, 71	sylibrd 262	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜)))
73	72	ralrimdva 3166	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜)))
74	1	kqfval 24097	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) = {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦})
75	74	adantr 486	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑧) = {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦})
76	1	kqfval 24097	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑤) = {𝑦 ∈ 𝐽 ∣ 𝑤 ∈ 𝑦})
77	76	adantlr 728	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑤) = {𝑦 ∈ 𝐽 ∣ 𝑤 ∈ 𝑦})
78	75, 77	eqeq12d 2779	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝑤 ∈ 𝑦}))
79		rabbi 3445	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) ↔ {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝑤 ∈ 𝑦})
80	49, 79	bitri 278	. . . . . . . . 9 ⊢ (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜) ↔ {𝑦 ∈ 𝐽 ∣ 𝑧 ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝑤 ∈ 𝑦})
81	78, 80	bitr4di 292	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))
82	81	biimprd 251	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜) → (𝐹‘𝑧) = (𝐹‘𝑤)))
83	73, 82	imim12d 82	. . . . . 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))))
84	83	ralimdva 3178	. . . . 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))))
85	84	ralimdva 3178	. . . 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))))
86		eleq1 2852	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 ∈ 𝑣 ↔ (𝐹‘𝑧) ∈ 𝑣))
87	86	imbi1d 344	. . . . . . . . . 10 ⊢ (𝑎 = (𝐹‘𝑧) → ((𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) ↔ ((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣)))
88	87	ralbidv 3189	. . . . . . . . 9 ⊢ (𝑎 = (𝐹‘𝑧) → (∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣)))
89		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑎 = (𝐹‘𝑧) → (𝑎 = 𝑏 ↔ (𝐹‘𝑧) = 𝑏))
90	88, 89	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)))
91	90	ralbidv 3189	. . . . . . 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)))
92	91	ralrn 7120	. . . . . 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)))
93		eleq1 2852	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐹‘𝑤) → (𝑏 ∈ 𝑣 ↔ (𝐹‘𝑤) ∈ 𝑣))
94	93	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑤) → (((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣) ↔ ((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣)))
95	94	ralbidv 3189	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑤) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → 𝑏 ∈ 𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑣 → (𝐹‘𝑤) ∈ 𝑣)))
96		eqeq2 2775	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑤) → ((𝐹‘𝑧) = 𝑏 ↔ (𝐹‘𝑧) = (𝐹‘𝑤)))
97	95, 96	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))))
98	97	ralrn 7120	. . . . . . 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))))
99	98	ralbidv 3189	. . . . . 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))))
100	92, 99	bitrd 282	. . . . 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))))
101	11, 100	syl 18	. . . 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))))
102	85, 101	sylibrd 262	. . 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)))
103		ist1-2 23720	. . . 4 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) → 𝑎 = 𝑏)))
104	33, 103	syl 18	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎 ∈ 𝑣 → 𝑏 ∈ 𝑣) → 𝑎 = 𝑏)))
105	102, 104	sylibrd 262	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)) → (KQ‘𝐽) ∈ Fre))
106	55, 105	impbid 215	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 {crab 3416 ∪ cuni 4872 ↦ cmpt 5192 ^◡ccnv 5657 dom cdm 5658 ran crn 5659 “ cima 5661 Fun wfun 6560 Fn wfn 6561 ‘cfv 6566 (class class class)co 7451 TopOnctopon 23283 Cn ccn 23597 Frect1 23680 KQckq 24067

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-map 8887 df-topgen 17664 df-qtop 17729 df-top 23267 df-topon 23284 df-cld 23392 df-cn 23600 df-t1 23687 df-kq 24068

This theorem is used by: r0sep 24122

W3C validator