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Theorem kqreglem1 21592
Description: A Kolmogorov quotient of a regular space is regular. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqreglem1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqreglem1
Dummy variables 𝑚 𝑤 𝑧 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . 5 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqtopon 21578 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
32adantr 480 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
4 topontop 20766 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
53, 4syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Top)
6 toponss 20779 . . . . . . . 8 (((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
73, 6sylan 487 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
87sselda 3636 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → 𝑏 ∈ ran 𝐹)
91kqffn 21576 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
109ad3antrrr 766 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → 𝐹 Fn 𝑋)
11 fvelrnb 6282 . . . . . . 7 (𝐹 Fn 𝑋 → (𝑏 ∈ ran 𝐹 ↔ ∃𝑧𝑋 (𝐹𝑧) = 𝑏))
1210, 11syl 17 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → (𝑏 ∈ ran 𝐹 ↔ ∃𝑧𝑋 (𝐹𝑧) = 𝑏))
138, 12mpbid 222 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → ∃𝑧𝑋 (𝐹𝑧) = 𝑏)
14 simpllr 815 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝐽 ∈ Reg)
151kqid 21579 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
1615ad3antrrr 766 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
17 simplr 807 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝑎 ∈ (KQ‘𝐽))
18 cnima 21117 . . . . . . . . . . . . . 14 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑎 ∈ (KQ‘𝐽)) → (𝐹𝑎) ∈ 𝐽)
1916, 17, 18syl2anc 694 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → (𝐹𝑎) ∈ 𝐽)
209adantr 480 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
2120adantr 480 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋)
22 elpreima 6377 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝑋 → (𝑧 ∈ (𝐹𝑎) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)))
2321, 22syl 17 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → (𝑧 ∈ (𝐹𝑎) ↔ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)))
2423biimpar 501 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝑧 ∈ (𝐹𝑎))
25 regsep 21186 . . . . . . . . . . . . 13 ((𝐽 ∈ Reg ∧ (𝐹𝑎) ∈ 𝐽𝑧 ∈ (𝐹𝑎)) → ∃𝑤𝐽 (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))
2614, 19, 24, 25syl3anc 1366 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → ∃𝑤𝐽 (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))
27 simp-4l 823 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝐽 ∈ (TopOn‘𝑋))
28 simprl 809 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑤𝐽)
291kqopn 21585 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) → (𝐹𝑤) ∈ (KQ‘𝐽))
3027, 28, 29syl2anc 694 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝐹𝑤) ∈ (KQ‘𝐽))
31 simprrl 821 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑧𝑤)
32 simplrl 817 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑧𝑋)
331kqfvima 21581 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽𝑧𝑋) → (𝑧𝑤 ↔ (𝐹𝑧) ∈ (𝐹𝑤)))
3427, 28, 32, 33syl3anc 1366 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝑧𝑤 ↔ (𝐹𝑧) ∈ (𝐹𝑤)))
3531, 34mpbid 222 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝐹𝑧) ∈ (𝐹𝑤))
36 topontop 20766 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3727, 36syl 17 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝐽 ∈ Top)
38 elssuni 4499 . . . . . . . . . . . . . . . . . 18 (𝑤𝐽𝑤 𝐽)
3938ad2antrl 764 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑤 𝐽)
40 eqid 2651 . . . . . . . . . . . . . . . . . 18 𝐽 = 𝐽
4140clscld 20899 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑤 𝐽) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽))
4237, 39, 41syl2anc 694 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽))
431kqcld 21586 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)))
4427, 42, 43syl2anc 694 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)))
4540sscls 20908 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑤 𝐽) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤))
4637, 39, 45syl2anc 694 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤))
47 imass2 5536 . . . . . . . . . . . . . . . 16 (𝑤 ⊆ ((cls‘𝐽)‘𝑤) → (𝐹𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
4846, 47syl 17 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝐹𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
49 eqid 2651 . . . . . . . . . . . . . . . 16 (KQ‘𝐽) = (KQ‘𝐽)
5049clsss2 20924 . . . . . . . . . . . . . . 15 (((𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
5144, 48, 50syl2anc 694 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
5220ad3antrrr 766 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝐹 Fn 𝑋)
53 fnfun 6026 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝑋 → Fun 𝐹)
5452, 53syl 17 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → Fun 𝐹)
55 simprrr 822 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎))
56 funimass2 6010 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎)
5754, 55, 56syl2anc 694 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎)
5851, 57sstrd 3646 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ 𝑎)
59 eleq2 2719 . . . . . . . . . . . . . . 15 (𝑚 = (𝐹𝑤) → ((𝐹𝑧) ∈ 𝑚 ↔ (𝐹𝑧) ∈ (𝐹𝑤)))
60 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑚 = (𝐹𝑤) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹𝑤)))
6160sseq1d 3665 . . . . . . . . . . . . . . 15 (𝑚 = (𝐹𝑤) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎 ↔ ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ 𝑎))
6259, 61anbi12d 747 . . . . . . . . . . . . . 14 (𝑚 = (𝐹𝑤) → (((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ ((𝐹𝑧) ∈ (𝐹𝑤) ∧ ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ 𝑎)))
6362rspcev 3340 . . . . . . . . . . . . 13 (((𝐹𝑤) ∈ (KQ‘𝐽) ∧ ((𝐹𝑧) ∈ (𝐹𝑤) ∧ ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
6430, 35, 58, 63syl12anc 1364 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
6526, 64rexlimddv 3064 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
6665expr 642 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
67 eleq1 2718 . . . . . . . . . . 11 ((𝐹𝑧) = 𝑏 → ((𝐹𝑧) ∈ 𝑎𝑏𝑎))
68 eleq1 2718 . . . . . . . . . . . . 13 ((𝐹𝑧) = 𝑏 → ((𝐹𝑧) ∈ 𝑚𝑏𝑚))
6968anbi1d 741 . . . . . . . . . . . 12 ((𝐹𝑧) = 𝑏 → (((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ (𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7069rexbidv 3081 . . . . . . . . . . 11 ((𝐹𝑧) = 𝑏 → (∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7167, 70imbi12d 333 . . . . . . . . . 10 ((𝐹𝑧) = 𝑏 → (((𝐹𝑧) ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)) ↔ (𝑏𝑎 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))))
7266, 71syl5ibcom 235 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧𝑋) → ((𝐹𝑧) = 𝑏 → (𝑏𝑎 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))))
7372com23 86 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧𝑋) → (𝑏𝑎 → ((𝐹𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))))
7473imp 444 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧𝑋) ∧ 𝑏𝑎) → ((𝐹𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7574an32s 863 . . . . . 6 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) ∧ 𝑧𝑋) → ((𝐹𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7675rexlimdva 3060 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → (∃𝑧𝑋 (𝐹𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7713, 76mpd 15 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
7877anasss 680 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑎 ∈ (KQ‘𝐽) ∧ 𝑏𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
7978ralrimivva 3000 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ (KQ‘𝐽)∀𝑏𝑎𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
80 isreg 21184 . 2 ((KQ‘𝐽) ∈ Reg ↔ ((KQ‘𝐽) ∈ Top ∧ ∀𝑎 ∈ (KQ‘𝐽)∀𝑏𝑎𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
815, 79, 80sylanbrc 699 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  wss 3607   cuni 4468  cmpt 4762  ccnv 5142  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  cfv 5926  (class class class)co 6690  Topctop 20746  TopOnctopon 20763  Clsdccld 20868  clsccl 20870   Cn ccn 21076  Regcreg 21161  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-int 4508  df-iun 4554  df-iin 4555  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-qtop 16214  df-top 20747  df-topon 20764  df-cld 20871  df-cls 20873  df-cn 21079  df-reg 21168  df-kq 21545
This theorem is referenced by:  kqreg  21602
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