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Theorem isr0 21450
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 21441 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
32ad2antrr 761 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
4 cnima 20979 . . . . . . . . . 10 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (𝐹𝑣) ∈ 𝐽)
53, 4sylan 488 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → (𝐹𝑣) ∈ 𝐽)
6 eleq2 2687 . . . . . . . . . . 11 (𝑜 = (𝐹𝑣) → (𝑧𝑜𝑧 ∈ (𝐹𝑣)))
7 eleq2 2687 . . . . . . . . . . 11 (𝑜 = (𝐹𝑣) → (𝑤𝑜𝑤 ∈ (𝐹𝑣)))
86, 7imbi12d 334 . . . . . . . . . 10 (𝑜 = (𝐹𝑣) → ((𝑧𝑜𝑤𝑜) ↔ (𝑧 ∈ (𝐹𝑣) → 𝑤 ∈ (𝐹𝑣))))
98rspcv 3291 . . . . . . . . 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 21438 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
1211ad2antrr 761 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝐹 Fn 𝑋)
1312adantr 481 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋)
14 fnfun 5946 . . . . . . . . . . 11 (𝐹 Fn 𝑋 → Fun 𝐹)
1513, 14syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → Fun 𝐹)
16 simprl 793 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝑧𝑋)
1716adantr 481 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧𝑋)
18 fndm 5948 . . . . . . . . . . . 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 2701 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑧 ∈ dom 𝐹)
21 fvimacnv 6288 . . . . . . . . . 10 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) ∈ 𝑣𝑧 ∈ (𝐹𝑣)))
2215, 20, 21syl2anc 692 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹𝑧) ∈ 𝑣𝑧 ∈ (𝐹𝑣)))
23 simprr 795 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → 𝑤𝑋)
2423adantr 481 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤𝑋)
2524, 19eleqtrrd 2701 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → 𝑤 ∈ dom 𝐹)
26 fvimacnv 6288 . . . . . . . . . 10 ((Fun 𝐹𝑤 ∈ dom 𝐹) → ((𝐹𝑤) ∈ 𝑣𝑤 ∈ (𝐹𝑣)))
2715, 25, 26syl2anc 692 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) ∧ 𝑣 ∈ (KQ‘𝐽)) → ((𝐹𝑤) ∈ 𝑣𝑤 ∈ (𝐹𝑣)))
2822, 27imbi12d 334 . . . . . . . 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 2963 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
31 simplr 791 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (KQ‘𝐽) ∈ Fre)
32 fnfvelrn 6312 . . . . . . . . 9 ((𝐹 Fn 𝑋𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
3312, 16, 32syl2anc 692 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑧) ∈ ran 𝐹)
341kqtopon 21440 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
3534ad2antrr 761 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
36 toponuni 20642 . . . . . . . . 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 2700 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑧) ∈ (KQ‘𝐽))
39 fnfvelrn 6312 . . . . . . . . 9 ((𝐹 Fn 𝑋𝑤𝑋) → (𝐹𝑤) ∈ ran 𝐹)
4012, 23, 39syl2anc 692 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑤) ∈ ran 𝐹)
4140, 37eleqtrd 2700 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹𝑤) ∈ (KQ‘𝐽))
42 eqid 2621 . . . . . . . 8 (KQ‘𝐽) = (KQ‘𝐽)
4342t1sep2 21083 . . . . . . 7 (((KQ‘𝐽) ∈ Fre ∧ (𝐹𝑧) ∈ (KQ‘𝐽) ∧ (𝐹𝑤) ∈ (KQ‘𝐽)) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤)))
4431, 38, 41, 43syl3anc 1323 . . . . . 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 21437 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
47 eleq2 2687 . . . . . . . . . 10 (𝑜 = 𝑦 → (𝑧𝑜𝑧𝑦))
48 eleq2 2687 . . . . . . . . . 10 (𝑜 = 𝑦 → (𝑤𝑜𝑤𝑦))
4947, 48bibi12d 335 . . . . . . . . 9 (𝑜 = 𝑦 → ((𝑧𝑜𝑤𝑜) ↔ (𝑧𝑦𝑤𝑦)))
5049cbvralv 3159 . . . . . . . 8 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦))
5146, 50syl6bbr 278 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
52513expb 1263 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5352adantlr 750 . . . . 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 2965 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
5655ex 450 . 2 (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre → ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜))))
57 simpll 789 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → 𝐽 ∈ (TopOn‘𝑋))
581kqopn 21447 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) → (𝐹𝑜) ∈ (KQ‘𝐽))
5957, 58sylan 488 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝐹𝑜) ∈ (KQ‘𝐽))
60 eleq2 2687 . . . . . . . . . . . 12 (𝑣 = (𝐹𝑜) → ((𝐹𝑧) ∈ 𝑣 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
61 eleq2 2687 . . . . . . . . . . . 12 (𝑣 = (𝐹𝑜) → ((𝐹𝑤) ∈ 𝑣 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
6260, 61imbi12d 334 . . . . . . . . . . 11 (𝑣 = (𝐹𝑜) → (((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) ↔ ((𝐹𝑧) ∈ (𝐹𝑜) → (𝐹𝑤) ∈ (𝐹𝑜))))
6362rspcv 3291 . . . . . . . . . 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 21443 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽𝑧𝑋) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
66653expa 1262 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) ∧ 𝑧𝑋) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
6766an32s 845 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑜𝐽) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
6867adantlr 750 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑧𝑜 ↔ (𝐹𝑧) ∈ (𝐹𝑜)))
691kqfvima 21443 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽𝑤𝑋) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
70693expa 1262 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) ∧ 𝑤𝑋) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
7170an32s 845 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
7271adantllr 754 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) ∧ 𝑜𝐽) → (𝑤𝑜 ↔ (𝐹𝑤) ∈ (𝐹𝑜)))
7368, 72imbi12d 334 . . . . . . . . 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 2963 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)))
761kqfval 21436 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) → (𝐹𝑧) = {𝑦𝐽𝑧𝑦})
7776adantr 481 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (𝐹𝑧) = {𝑦𝐽𝑧𝑦})
781kqfval 21436 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝑋) → (𝐹𝑤) = {𝑦𝐽𝑤𝑦})
7978adantlr 750 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → (𝐹𝑤) = {𝑦𝐽𝑤𝑦})
8077, 79eqeq12d 2636 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ {𝑦𝐽𝑧𝑦} = {𝑦𝐽𝑤𝑦}))
81 rabbi 3109 . . . . . . . . . 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 2956 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋) → (∀𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → ∀𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
8786ralimdva 2956 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜)) → ∀𝑧𝑋𝑤𝑋 (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
88 eleq1 2686 . . . . . . . . . . 11 (𝑎 = (𝐹𝑧) → (𝑎𝑣 ↔ (𝐹𝑧) ∈ 𝑣))
8988imbi1d 331 . . . . . . . . . 10 (𝑎 = (𝐹𝑧) → ((𝑎𝑣𝑏𝑣) ↔ ((𝐹𝑧) ∈ 𝑣𝑏𝑣)))
9089ralbidv 2980 . . . . . . . . 9 (𝑎 = (𝐹𝑧) → (∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣)))
91 eqeq1 2625 . . . . . . . . 9 (𝑎 = (𝐹𝑧) → (𝑎 = 𝑏 ↔ (𝐹𝑧) = 𝑏))
9290, 91imbi12d 334 . . . . . . . 8 (𝑎 = (𝐹𝑧) → ((∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏) ↔ (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏)))
9392ralbidv 2980 . . . . . . 7 (𝑎 = (𝐹𝑧) → (∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)(𝑎𝑣𝑏𝑣) → 𝑎 = 𝑏) ↔ ∀𝑏 ∈ ran 𝐹(∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏)))
9493ralrn 6318 . . . . . 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 2686 . . . . . . . . . . 11 (𝑏 = (𝐹𝑤) → (𝑏𝑣 ↔ (𝐹𝑤) ∈ 𝑣))
9695imbi2d 330 . . . . . . . . . 10 (𝑏 = (𝐹𝑤) → (((𝐹𝑧) ∈ 𝑣𝑏𝑣) ↔ ((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
9796ralbidv 2980 . . . . . . . . 9 (𝑏 = (𝐹𝑤) → (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) ↔ ∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣)))
98 eqeq2 2632 . . . . . . . . 9 (𝑏 = (𝐹𝑤) → ((𝐹𝑧) = 𝑏 ↔ (𝐹𝑧) = (𝐹𝑤)))
9997, 98imbi12d 334 . . . . . . . 8 (𝑏 = (𝐹𝑤) → ((∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣𝑏𝑣) → (𝐹𝑧) = 𝑏) ↔ (∀𝑣 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑣 → (𝐹𝑤) ∈ 𝑣) → (𝐹𝑧) = (𝐹𝑤))))
10099ralrn 6318 . . . . . . 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 2980 . . . . . 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 21061 . . . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911   cuni 4402  cmpt 4673  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  Fun wfun 5841   Fn wfn 5842  cfv 5847  (class class class)co 6604  TopOnctopon 20618   Cn ccn 20938  Frect1 21021  KQckq 21406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-topgen 16025  df-qtop 16088  df-top 20621  df-topon 20623  df-cld 20733  df-cn 20941  df-t1 21028  df-kq 21407
This theorem is referenced by:  r0sep  21461
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