MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isr0 Structured version   Visualization version   GIF version

Theorem isr0 23761
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 23752 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
32ad2antrr 726 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
4 cnima 23289 . . . . . . . . . 10 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (𝐹𝑣) ∈ 𝐽)
53, 4sylan 580 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (𝐹𝑣) ∈ 𝐽)
6 eleq2 2828 . . . . . . . . . . 11 (𝑜 = (𝐹𝑣) → (𝑧𝑜𝑧 ∈ (𝐹𝑣)))
7 eleq2 2828 . . . . . . . . . . 11 (𝑜 = (𝐹𝑣) → (𝑤𝑜𝑤 ∈ (𝐹𝑣)))
86, 7imbi12d 344 . . . . . . . . . 10 (𝑜 = (𝐹𝑣) → ((𝑧𝑜𝑤𝑜) ↔ (𝑧 ∈ (𝐹𝑣) → 𝑤 ∈ (𝐹𝑣))))
98rspcv 3618 . . . . . . . . 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 23749 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
1211ad2antrr 726 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝐹 Fn 𝑋)
1312adantr 480 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋)
14 fnfun 6669 . . . . . . . . . . 11 (𝐹 Fn 𝑋 → Fun 𝐹)
1513, 14syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → Fun 𝐹)
16 simprl 771 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝑧𝑋)
1716adantr 480 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧𝑋)
1813fndmd 6674 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → dom 𝐹 = 𝑋)
1917, 18eleqtrrd 2842 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧 ∈ dom 𝐹)
20 fvimacnv 7073 . . . . . . . . . 10 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) ∈ 𝑣𝑧 ∈ (𝐹𝑣)))
2115, 19, 20syl2anc 584 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹𝑧) ∈ 𝑣𝑧 ∈ (𝐹𝑣)))
22 simprr 773 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝑤𝑋)
2322adantr 480 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤𝑋)
2423, 18eleqtrrd 2842 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤 ∈ dom 𝐹)
25 fvimacnv 7073 . . . . . . . . . 10 ((Fun 𝐹𝑤 ∈ dom 𝐹) → ((𝐹𝑤) ∈ 𝑣𝑤 ∈ (𝐹𝑣)))
2615, 24, 25syl2anc 584 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹𝑤) ∈ 𝑣𝑤 ∈ (𝐹𝑣)))
2721, 26imbi12d 344 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) ↔ (𝑧 ∈ (𝐹𝑣) → 𝑤 ∈ (𝐹𝑣))))
2810, 27sylibrd 259 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
2928ralrimdva 3152 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
30 simplr 769 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (KQ‘𝐽) ∈ Fre)
31 fnfvelrn 7100 . . . . . . . . 9 ((𝐹 Fn 𝑋𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
3212, 16, 31syl2anc 584 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑧) ∈ ran 𝐹)
331kqtopon 23751 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
3433ad2antrr 726 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
35 toponuni 22936 . . . . . . . . 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 2841 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑧) ∈ (KQ‘𝐽))
38 fnfvelrn 7100 . . . . . . . . 9 ((𝐹 Fn 𝑋𝑤𝑋) → (𝐹𝑤) ∈ ran 𝐹)
3912, 22, 38syl2anc 584 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑤) ∈ ran 𝐹)
4039, 36eleqtrd 2841 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑤) ∈ (KQ‘𝐽))
41 eqid 2735 . . . . . . . 8 (KQ‘𝐽) = (KQ‘𝐽)
4241t1sep2 23393 . . . . . . 7 (((KQ‘𝐽) ∈ Fre ∧ (𝐹𝑧) ∈ (KQ‘𝐽) ∧ (𝐹𝑤) ∈ (KQ‘𝐽)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤)))
4330, 37, 40, 42syl3anc 1370 . . . . . 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 23748 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
46 eleq2 2828 . . . . . . . . . 10 (𝑜 = 𝑦 → (𝑧𝑜𝑧𝑦))
47 eleq2 2828 . . . . . . . . . 10 (𝑜 = 𝑦 → (𝑤𝑜𝑤𝑦))
4846, 47bibi12d 345 . . . . . . . . 9 (𝑜 = 𝑦 → ((𝑧𝑜𝑤𝑜) ↔ (𝑧𝑦𝑤𝑦)))
4948cbvralvw 3235 . . . . . . . 8 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦))
5045, 49bitr4di 289 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
51503expb 1119 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5251adantlr 715 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5344, 52sylibd 239 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5453ralrimivva 3200 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5554ex 412 . 2 (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre → ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜))))
561kqopn 23758 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) → (𝐹𝑜) ∈ (KQ‘𝐽))
5756ad4ant14 752 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝐹𝑜) ∈ (KQ‘𝐽))
58 eleq2 2828 . . . . . . . . . . . 12 (𝑣 = (𝐹𝑜) → ((𝐹𝑧) ∈ 𝑣 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
59 eleq2 2828 . . . . . . . . . . . 12 (𝑣 = (𝐹𝑜) → ((𝐹𝑤) ∈ 𝑣 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
6058, 59imbi12d 344 . . . . . . . . . . 11 (𝑣 = (𝐹𝑜) → (((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) ↔ ((𝐹𝑧) ∈ (𝐹𝑜) → (𝐹𝑤) ∈ (𝐹𝑜))))
6160rspcv 3618 . . . . . . . . . 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 23754 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽𝑧𝑋) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
64633expa 1117 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) ∧ 𝑧𝑋) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
6564an32s 652 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑜𝐽) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
6665adantlr 715 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
671kqfvima 23754 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽𝑤𝑋) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
68673expa 1117 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) ∧ 𝑤𝑋) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
6968an32s 652 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
7069adantllr 719 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
7166, 70imbi12d 344 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → ((𝑧𝑜𝑤𝑜) ↔ ((𝐹𝑧) ∈ (𝐹𝑜) → (𝐹𝑤) ∈ (𝐹𝑜))))
7262, 71sylibrd 259 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝑧𝑜𝑤𝑜)))
7372ralrimdva 3152 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
741kqfval 23747 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) → (𝐹𝑧) = {𝑦𝐽𝑧𝑦})
7574adantr 480 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (𝐹𝑧) = {𝑦𝐽𝑧𝑦})
761kqfval 23747 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝑋) → (𝐹𝑤) = {𝑦𝐽𝑤𝑦})
7776adantlr 715 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (𝐹𝑤) = {𝑦𝐽𝑤𝑦})
7875, 77eqeq12d 2751 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ {𝑦𝐽𝑧𝑦} = {𝑦𝐽𝑤𝑦}))
79 rabbi 3465 . . . . . . . . . 10 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) ↔ {𝑦𝐽𝑧𝑦} = {𝑦𝐽𝑤𝑦})
8049, 79bitri 275 . . . . . . . . 9 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) ↔ {𝑦𝐽𝑧𝑦} = {𝑦𝐽𝑤𝑦})
8178, 80bitr4di 289 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
8281biimprd 248 . . . . . . 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 3165 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) → (∀𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → ∀𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
8584ralimdva 3165 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → ∀𝑧𝑋𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
86 eleq1 2827 . . . . . . . . . . 11 (𝑎 = (𝐹𝑧) → (𝑎𝑣 ↔ (𝐹𝑧) ∈ 𝑣))
8786imbi1d 341 . . . . . . . . . 10 (𝑎 = (𝐹𝑧) → ((𝑎𝑣𝑏𝑣) ↔ ((𝐹𝑧) ∈ 𝑣𝑏𝑣)))
8887ralbidv 3176 . . . . . . . . 9 (𝑎 = (𝐹𝑧) → (∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣)))
89 eqeq1 2739 . . . . . . . . 9 (𝑎 = (𝐹𝑧) → (𝑎 = 𝑏 ↔ (𝐹𝑧) = 𝑏))
9088, 89imbi12d 344 . . . . . . . 8 (𝑎 = (𝐹𝑧) → ((∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏) ↔ (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏)))
9190ralbidv 3176 . . . . . . 7 (𝑎 = (𝐹𝑧) → (∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏) ↔ ∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏)))
9291ralrn 7108 . . . . . 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 2827 . . . . . . . . . . 11 (𝑏 = (𝐹𝑤) → (𝑏𝑣 ↔ (𝐹𝑤) ∈ 𝑣))
9493imbi2d 340 . . . . . . . . . 10 (𝑏 = (𝐹𝑤) → (((𝐹𝑧) ∈ 𝑣𝑏𝑣) ↔ ((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
9594ralbidv 3176 . . . . . . . . 9 (𝑏 = (𝐹𝑤) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
96 eqeq2 2747 . . . . . . . . 9 (𝑏 = (𝐹𝑤) → ((𝐹𝑧) = 𝑏 ↔ (𝐹𝑧) = (𝐹𝑤)))
9795, 96imbi12d 344 . . . . . . . 8 (𝑏 = (𝐹𝑤) → ((∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏) ↔ (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
9897ralrn 7108 . . . . . . 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 3176 . . . . . 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 279 . . . . 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 259 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏)))
103 ist1-2 23371 . . . 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 259 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → (KQ‘𝐽) ∈ Fre))
10655, 105impbid 212 1 (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433   cuni 4912  cmpt 5231  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557   Fn wfn 6558  cfv 6563  (class class class)co 7431  TopOnctopon 22932   Cn ccn 23248  Frect1 23331  KQckq 23717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-topgen 17490  df-qtop 17554  df-top 22916  df-topon 22933  df-cld 23043  df-cn 23251  df-t1 23338  df-kq 23718
This theorem is referenced by:  r0sep  23772
  Copyright terms: Public domain W3C validator