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Theorem isr0 23685
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 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
isr0 (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜))))
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.
StepHypRef Expression
1 kqval.2 . . . . . . . . . . . 12 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqid 23676 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
32ad2antrr 724 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
4 cnima 23213 . . . . . . . . . 10 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (𝐹𝑣) ∈ 𝐽)
53, 4sylan 578 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (𝐹𝑣) ∈ 𝐽)
6 eleq2 2814 . . . . . . . . . . 11 (𝑜 = (𝐹𝑣) → (𝑧𝑜𝑧 ∈ (𝐹𝑣)))
7 eleq2 2814 . . . . . . . . . . 11 (𝑜 = (𝐹𝑣) → (𝑤𝑜𝑤 ∈ (𝐹𝑣)))
86, 7imbi12d 343 . . . . . . . . . 10 (𝑜 = (𝐹𝑣) → ((𝑧𝑜𝑤𝑜) ↔ (𝑧 ∈ (𝐹𝑣) → 𝑤 ∈ (𝐹𝑣))))
98rspcv 3602 . . . . . . . . 9 ((𝐹𝑣) ∈ 𝐽 → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → (𝑧 ∈ (𝐹𝑣) → 𝑤 ∈ (𝐹𝑣))))
105, 9syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → (𝑧 ∈ (𝐹𝑣) → 𝑤 ∈ (𝐹𝑣))))
111kqffn 23673 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
1211ad2antrr 724 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝐹 Fn 𝑋)
1312adantr 479 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋)
14 fnfun 6655 . . . . . . . . . . 11 (𝐹 Fn 𝑋 → Fun 𝐹)
1513, 14syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → Fun 𝐹)
16 simprl 769 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝑧𝑋)
1716adantr 479 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧𝑋)
1813fndmd 6660 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → dom 𝐹 = 𝑋)
1917, 18eleqtrrd 2828 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧 ∈ dom 𝐹)
20 fvimacnv 7061 . . . . . . . . . 10 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) ∈ 𝑣𝑧 ∈ (𝐹𝑣)))
2115, 19, 20syl2anc 582 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹𝑧) ∈ 𝑣𝑧 ∈ (𝐹𝑣)))
22 simprr 771 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝑤𝑋)
2322adantr 479 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤𝑋)
2423, 18eleqtrrd 2828 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤 ∈ dom 𝐹)
25 fvimacnv 7061 . . . . . . . . . 10 ((Fun 𝐹𝑤 ∈ dom 𝐹) → ((𝐹𝑤) ∈ 𝑣𝑤 ∈ (𝐹𝑣)))
2615, 24, 25syl2anc 582 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹𝑤) ∈ 𝑣𝑤 ∈ (𝐹𝑣)))
2721, 26imbi12d 343 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) ↔ (𝑧 ∈ (𝐹𝑣) → 𝑤 ∈ (𝐹𝑣))))
2810, 27sylibrd 258 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
2928ralrimdva 3143 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
30 simplr 767 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (KQ‘𝐽) ∈ Fre)
31 fnfvelrn 7089 . . . . . . . . 9 ((𝐹 Fn 𝑋𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
3212, 16, 31syl2anc 582 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑧) ∈ ran 𝐹)
331kqtopon 23675 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
3433ad2antrr 724 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
35 toponuni 22860 . . . . . . . . 9 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = (KQ‘𝐽))
3634, 35syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → ran 𝐹 = (KQ‘𝐽))
3732, 36eleqtrd 2827 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑧) ∈ (KQ‘𝐽))
38 fnfvelrn 7089 . . . . . . . . 9 ((𝐹 Fn 𝑋𝑤𝑋) → (𝐹𝑤) ∈ ran 𝐹)
3912, 22, 38syl2anc 582 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑤) ∈ ran 𝐹)
4039, 36eleqtrd 2827 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑤) ∈ (KQ‘𝐽))
41 eqid 2725 . . . . . . . 8 (KQ‘𝐽) = (KQ‘𝐽)
4241t1sep2 23317 . . . . . . 7 (((KQ‘𝐽) ∈ Fre ∧ (𝐹𝑧) ∈ (KQ‘𝐽) ∧ (𝐹𝑤) ∈ (KQ‘𝐽)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤)))
4330, 37, 40, 42syl3anc 1368 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤)))
4429, 43syld 47 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → (𝐹𝑧) = (𝐹𝑤)))
451kqfeq 23672 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
46 eleq2 2814 . . . . . . . . . 10 (𝑜 = 𝑦 → (𝑧𝑜𝑧𝑦))
47 eleq2 2814 . . . . . . . . . 10 (𝑜 = 𝑦 → (𝑤𝑜𝑤𝑦))
4846, 47bibi12d 344 . . . . . . . . 9 (𝑜 = 𝑦 → ((𝑧𝑜𝑤𝑜) ↔ (𝑧𝑦𝑤𝑦)))
4948cbvralvw 3224 . . . . . . . 8 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦))
5045, 49bitr4di 288 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
51503expb 1117 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5251adantlr 713 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5344, 52sylibd 238 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5453ralrimivva 3190 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5554ex 411 . 2 (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre → ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜))))
561kqopn 23682 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) → (𝐹𝑜) ∈ (KQ‘𝐽))
5756ad4ant14 750 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝐹𝑜) ∈ (KQ‘𝐽))
58 eleq2 2814 . . . . . . . . . . . 12 (𝑣 = (𝐹𝑜) → ((𝐹𝑧) ∈ 𝑣 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
59 eleq2 2814 . . . . . . . . . . . 12 (𝑣 = (𝐹𝑜) → ((𝐹𝑤) ∈ 𝑣 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
6058, 59imbi12d 343 . . . . . . . . . . 11 (𝑣 = (𝐹𝑜) → (((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) ↔ ((𝐹𝑧) ∈ (𝐹𝑜) → (𝐹𝑤) ∈ (𝐹𝑜))))
6160rspcv 3602 . . . . . . . . . 10 ((𝐹𝑜) ∈ (KQ‘𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → ((𝐹𝑧) ∈ (𝐹𝑜) → (𝐹𝑤) ∈ (𝐹𝑜))))
6257, 61syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → ((𝐹𝑧) ∈ (𝐹𝑜) → (𝐹𝑤) ∈ (𝐹𝑜))))
631kqfvima 23678 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽𝑧𝑋) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
64633expa 1115 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) ∧ 𝑧𝑋) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
6564an32s 650 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑜𝐽) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
6665adantlr 713 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
671kqfvima 23678 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽𝑤𝑋) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
68673expa 1115 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) ∧ 𝑤𝑋) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
6968an32s 650 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
7069adantllr 717 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
7166, 70imbi12d 343 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → ((𝑧𝑜𝑤𝑜) ↔ ((𝐹𝑧) ∈ (𝐹𝑜) → (𝐹𝑤) ∈ (𝐹𝑜))))
7262, 71sylibrd 258 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝑧𝑜𝑤𝑜)))
7372ralrimdva 3143 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
741kqfval 23671 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) → (𝐹𝑧) = {𝑦𝐽𝑧𝑦})
7574adantr 479 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (𝐹𝑧) = {𝑦𝐽𝑧𝑦})
761kqfval 23671 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝑋) → (𝐹𝑤) = {𝑦𝐽𝑤𝑦})
7776adantlr 713 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (𝐹𝑤) = {𝑦𝐽𝑤𝑦})
7875, 77eqeq12d 2741 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ {𝑦𝐽𝑧𝑦} = {𝑦𝐽𝑤𝑦}))
79 rabbi 3449 . . . . . . . . . 10 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) ↔ {𝑦𝐽𝑧𝑦} = {𝑦𝐽𝑤𝑦})
8049, 79bitri 274 . . . . . . . . 9 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) ↔ {𝑦𝐽𝑧𝑦} = {𝑦𝐽𝑤𝑦})
8178, 80bitr4di 288 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
8281biimprd 247 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → (𝐹𝑧) = (𝐹𝑤)))
8373, 82imim12d 81 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
8483ralimdva 3156 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) → (∀𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → ∀𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
8584ralimdva 3156 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → ∀𝑧𝑋𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
86 eleq1 2813 . . . . . . . . . . 11 (𝑎 = (𝐹𝑧) → (𝑎𝑣 ↔ (𝐹𝑧) ∈ 𝑣))
8786imbi1d 340 . . . . . . . . . 10 (𝑎 = (𝐹𝑧) → ((𝑎𝑣𝑏𝑣) ↔ ((𝐹𝑧) ∈ 𝑣𝑏𝑣)))
8887ralbidv 3167 . . . . . . . . 9 (𝑎 = (𝐹𝑧) → (∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣)))
89 eqeq1 2729 . . . . . . . . 9 (𝑎 = (𝐹𝑧) → (𝑎 = 𝑏 ↔ (𝐹𝑧) = 𝑏))
9088, 89imbi12d 343 . . . . . . . 8 (𝑎 = (𝐹𝑧) → ((∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏) ↔ (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏)))
9190ralbidv 3167 . . . . . . 7 (𝑎 = (𝐹𝑧) → (∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏) ↔ ∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏)))
9291ralrn 7097 . . . . . 6 (𝐹 Fn 𝑋 → (∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏) ↔ ∀𝑧𝑋𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏)))
93 eleq1 2813 . . . . . . . . . . 11 (𝑏 = (𝐹𝑤) → (𝑏𝑣 ↔ (𝐹𝑤) ∈ 𝑣))
9493imbi2d 339 . . . . . . . . . 10 (𝑏 = (𝐹𝑤) → (((𝐹𝑧) ∈ 𝑣𝑏𝑣) ↔ ((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
9594ralbidv 3167 . . . . . . . . 9 (𝑏 = (𝐹𝑤) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
96 eqeq2 2737 . . . . . . . . 9 (𝑏 = (𝐹𝑤) → ((𝐹𝑧) = 𝑏 ↔ (𝐹𝑧) = (𝐹𝑤)))
9795, 96imbi12d 343 . . . . . . . 8 (𝑏 = (𝐹𝑤) → ((∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏) ↔ (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
9897ralrn 7097 . . . . . . 7 (𝐹 Fn 𝑋 → (∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏) ↔ ∀𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
9998ralbidv 3167 . . . . . 6 (𝐹 Fn 𝑋 → (∀𝑧𝑋𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏) ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
10092, 99bitrd 278 . . . . 5 (𝐹 Fn 𝑋 → (∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏) ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
10111, 100syl 17 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏) ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
10285, 101sylibrd 258 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏)))
103 ist1-2 23295 . . . 4 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏)))
10433, 103syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏)))
105102, 104sylibrd 258 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → (KQ‘𝐽) ∈ Fre))
10655, 105impbid 211 1 (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  {crab 3418   cuni 4909  cmpt 5232  ccnv 5677  dom cdm 5678  ran crn 5679  cima 5681  Fun wfun 6543   Fn wfn 6544  cfv 6549  (class class class)co 7419  TopOnctopon 22856   Cn ccn 23172  Frect1 23255  KQckq 23641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-topgen 17428  df-qtop 17492  df-top 22840  df-topon 22857  df-cld 22967  df-cn 23175  df-t1 23262  df-kq 23642
This theorem is referenced by:  r0sep  23696
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