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Theorem isr0 21588
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 21579 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
32ad2antrr 762 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
4 cnima 21117 . . . . . . . . . 10 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (𝐹𝑣) ∈ 𝐽)
53, 4sylan 487 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (𝐹𝑣) ∈ 𝐽)
6 eleq2 2719 . . . . . . . . . . 11 (𝑜 = (𝐹𝑣) → (𝑧𝑜𝑧 ∈ (𝐹𝑣)))
7 eleq2 2719 . . . . . . . . . . 11 (𝑜 = (𝐹𝑣) → (𝑤𝑜𝑤 ∈ (𝐹𝑣)))
86, 7imbi12d 333 . . . . . . . . . 10 (𝑜 = (𝐹𝑣) → ((𝑧𝑜𝑤𝑜) ↔ (𝑧 ∈ (𝐹𝑣) → 𝑤 ∈ (𝐹𝑣))))
98rspcv 3336 . . . . . . . . 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 21576 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
1211ad2antrr 762 . . . . . . . . . . . 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 6026 . . . . . . . . . . 11 (𝐹 Fn 𝑋 → Fun 𝐹)
1513, 14syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → Fun 𝐹)
16 simprl 809 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝑧𝑋)
1716adantr 480 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧𝑋)
18 fndm 6028 . . . . . . . . . . . 12 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
1913, 18syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → dom 𝐹 = 𝑋)
2017, 19eleqtrrd 2733 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧 ∈ dom 𝐹)
21 fvimacnv 6372 . . . . . . . . . 10 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) ∈ 𝑣𝑧 ∈ (𝐹𝑣)))
2215, 20, 21syl2anc 694 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹𝑧) ∈ 𝑣𝑧 ∈ (𝐹𝑣)))
23 simprr 811 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝑤𝑋)
2423adantr 480 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤𝑋)
2524, 19eleqtrrd 2733 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤 ∈ dom 𝐹)
26 fvimacnv 6372 . . . . . . . . . 10 ((Fun 𝐹𝑤 ∈ dom 𝐹) → ((𝐹𝑤) ∈ 𝑣𝑤 ∈ (𝐹𝑣)))
2715, 25, 26syl2anc 694 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹𝑤) ∈ 𝑣𝑤 ∈ (𝐹𝑣)))
2822, 27imbi12d 333 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) ↔ (𝑧 ∈ (𝐹𝑣) → 𝑤 ∈ (𝐹𝑣))))
2910, 28sylibrd 249 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
3029ralrimdva 2998 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
31 simplr 807 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (KQ‘𝐽) ∈ Fre)
32 fnfvelrn 6396 . . . . . . . . 9 ((𝐹 Fn 𝑋𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
3312, 16, 32syl2anc 694 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑧) ∈ ran 𝐹)
341kqtopon 21578 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
3534ad2antrr 762 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
36 toponuni 20767 . . . . . . . . 9 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = (KQ‘𝐽))
3735, 36syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → ran 𝐹 = (KQ‘𝐽))
3833, 37eleqtrd 2732 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑧) ∈ (KQ‘𝐽))
39 fnfvelrn 6396 . . . . . . . . 9 ((𝐹 Fn 𝑋𝑤𝑋) → (𝐹𝑤) ∈ ran 𝐹)
4012, 23, 39syl2anc 694 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑤) ∈ ran 𝐹)
4140, 37eleqtrd 2732 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑤) ∈ (KQ‘𝐽))
42 eqid 2651 . . . . . . . 8 (KQ‘𝐽) = (KQ‘𝐽)
4342t1sep2 21221 . . . . . . 7 (((KQ‘𝐽) ∈ Fre ∧ (𝐹𝑧) ∈ (KQ‘𝐽) ∧ (𝐹𝑤) ∈ (KQ‘𝐽)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤)))
4431, 38, 41, 43syl3anc 1366 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤)))
4530, 44syld 47 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → (𝐹𝑧) = (𝐹𝑤)))
461kqfeq 21575 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
47 eleq2 2719 . . . . . . . . . 10 (𝑜 = 𝑦 → (𝑧𝑜𝑧𝑦))
48 eleq2 2719 . . . . . . . . . 10 (𝑜 = 𝑦 → (𝑤𝑜𝑤𝑦))
4947, 48bibi12d 334 . . . . . . . . 9 (𝑜 = 𝑦 → ((𝑧𝑜𝑤𝑜) ↔ (𝑧𝑦𝑤𝑦)))
5049cbvralv 3201 . . . . . . . 8 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦))
5146, 50syl6bbr 278 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
52513expb 1285 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5352adantlr 751 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5445, 53sylibd 229 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5554ralrimivva 3000 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5655ex 449 . 2 (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre → ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜))))
57 simpll 805 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → 𝐽 ∈ (TopOn‘𝑋))
581kqopn 21585 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) → (𝐹𝑜) ∈ (KQ‘𝐽))
5957, 58sylan 487 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝐹𝑜) ∈ (KQ‘𝐽))
60 eleq2 2719 . . . . . . . . . . . 12 (𝑣 = (𝐹𝑜) → ((𝐹𝑧) ∈ 𝑣 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
61 eleq2 2719 . . . . . . . . . . . 12 (𝑣 = (𝐹𝑜) → ((𝐹𝑤) ∈ 𝑣 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
6260, 61imbi12d 333 . . . . . . . . . . 11 (𝑣 = (𝐹𝑜) → (((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) ↔ ((𝐹𝑧) ∈ (𝐹𝑜) → (𝐹𝑤) ∈ (𝐹𝑜))))
6362rspcv 3336 . . . . . . . . . 10 ((𝐹𝑜) ∈ (KQ‘𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → ((𝐹𝑧) ∈ (𝐹𝑜) → (𝐹𝑤) ∈ (𝐹𝑜))))
6459, 63syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → ((𝐹𝑧) ∈ (𝐹𝑜) → (𝐹𝑤) ∈ (𝐹𝑜))))
651kqfvima 21581 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽𝑧𝑋) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
66653expa 1284 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) ∧ 𝑧𝑋) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
6766an32s 863 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑜𝐽) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
6867adantlr 751 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
691kqfvima 21581 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽𝑤𝑋) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
70693expa 1284 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) ∧ 𝑤𝑋) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
7170an32s 863 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
7271adantllr 755 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
7368, 72imbi12d 333 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → ((𝑧𝑜𝑤𝑜) ↔ ((𝐹𝑧) ∈ (𝐹𝑜) → (𝐹𝑤) ∈ (𝐹𝑜))))
7464, 73sylibrd 249 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝑧𝑜𝑤𝑜)))
7574ralrimdva 2998 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
761kqfval 21574 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) → (𝐹𝑧) = {𝑦𝐽𝑧𝑦})
7776adantr 480 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (𝐹𝑧) = {𝑦𝐽𝑧𝑦})
781kqfval 21574 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝑋) → (𝐹𝑤) = {𝑦𝐽𝑤𝑦})
7978adantlr 751 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (𝐹𝑤) = {𝑦𝐽𝑤𝑦})
8077, 79eqeq12d 2666 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ {𝑦𝐽𝑧𝑦} = {𝑦𝐽𝑤𝑦}))
81 rabbi 3150 . . . . . . . . . 10 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) ↔ {𝑦𝐽𝑧𝑦} = {𝑦𝐽𝑤𝑦})
8250, 81bitri 264 . . . . . . . . 9 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) ↔ {𝑦𝐽𝑧𝑦} = {𝑦𝐽𝑤𝑦})
8380, 82syl6bbr 278 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
8483biimprd 238 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → (𝐹𝑧) = (𝐹𝑤)))
8575, 84imim12d 81 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
8685ralimdva 2991 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) → (∀𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → ∀𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
8786ralimdva 2991 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → ∀𝑧𝑋𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
88 eleq1 2718 . . . . . . . . . . 11 (𝑎 = (𝐹𝑧) → (𝑎𝑣 ↔ (𝐹𝑧) ∈ 𝑣))
8988imbi1d 330 . . . . . . . . . 10 (𝑎 = (𝐹𝑧) → ((𝑎𝑣𝑏𝑣) ↔ ((𝐹𝑧) ∈ 𝑣𝑏𝑣)))
9089ralbidv 3015 . . . . . . . . 9 (𝑎 = (𝐹𝑧) → (∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣)))
91 eqeq1 2655 . . . . . . . . 9 (𝑎 = (𝐹𝑧) → (𝑎 = 𝑏 ↔ (𝐹𝑧) = 𝑏))
9290, 91imbi12d 333 . . . . . . . 8 (𝑎 = (𝐹𝑧) → ((∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏) ↔ (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏)))
9392ralbidv 3015 . . . . . . 7 (𝑎 = (𝐹𝑧) → (∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏) ↔ ∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏)))
9493ralrn 6402 . . . . . 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)))
95 eleq1 2718 . . . . . . . . . . 11 (𝑏 = (𝐹𝑤) → (𝑏𝑣 ↔ (𝐹𝑤) ∈ 𝑣))
9695imbi2d 329 . . . . . . . . . 10 (𝑏 = (𝐹𝑤) → (((𝐹𝑧) ∈ 𝑣𝑏𝑣) ↔ ((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
9796ralbidv 3015 . . . . . . . . 9 (𝑏 = (𝐹𝑤) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
98 eqeq2 2662 . . . . . . . . 9 (𝑏 = (𝐹𝑤) → ((𝐹𝑧) = 𝑏 ↔ (𝐹𝑧) = (𝐹𝑤)))
9997, 98imbi12d 333 . . . . . . . 8 (𝑏 = (𝐹𝑤) → ((∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏) ↔ (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
10099ralrn 6402 . . . . . . 7 (𝐹 Fn 𝑋 → (∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏) ↔ ∀𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
101100ralbidv 3015 . . . . . 6 (𝐹 Fn 𝑋 → (∀𝑧𝑋𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏) ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
10294, 101bitrd 268 . . . . 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))))
10311, 102syl 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))))
10487, 103sylibrd 249 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏)))
105 ist1-2 21199 . . . 4 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏)))
10634, 105syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏)))
107104, 106sylibrd 249 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → (KQ‘𝐽) ∈ Fre))
10856, 107impbid 202 1 (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945   cuni 4468  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  cfv 5926  (class class class)co 6690  TopOnctopon 20763   Cn ccn 21076  Frect1 21159  KQckq 21544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-topgen 16151  df-qtop 16214  df-top 20747  df-topon 20764  df-cld 20871  df-cn 21079  df-t1 21166  df-kq 21545
This theorem is referenced by:  r0sep  21599
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