MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqreglem1 Structured version   Visualization version   GIF version

Theorem kqreglem1 21454
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 21440 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
32adantr 481 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
4 topontop 20641 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
53, 4syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Top)
6 toponss 20644 . . . . . . . 8 (((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
73, 6sylan 488 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹)
87sselda 3583 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → 𝑏 ∈ ran 𝐹)
91kqffn 21438 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
109ad3antrrr 765 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) → 𝐹 Fn 𝑋)
11 fvelrnb 6200 . . . . . . 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 798 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝐽 ∈ Reg)
151kqid 21441 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
1615ad3antrrr 765 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
17 simplr 791 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝑎 ∈ (KQ‘𝐽))
18 cnima 20979 . . . . . . . . . . . . . 14 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑎 ∈ (KQ‘𝐽)) → (𝐹𝑎) ∈ 𝐽)
1916, 17, 18syl2anc 692 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → (𝐹𝑎) ∈ 𝐽)
209adantr 481 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋)
2120adantr 481 . . . . . . . . . . . . . . 15 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋)
22 elpreima 6293 . . . . . . . . . . . . . . 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 502 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → 𝑧 ∈ (𝐹𝑎))
25 regsep 21048 . . . . . . . . . . . . 13 ((𝐽 ∈ Reg ∧ (𝐹𝑎) ∈ 𝐽𝑧 ∈ (𝐹𝑎)) → ∃𝑤𝐽 (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))
2614, 19, 24, 25syl3anc 1323 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) → ∃𝑤𝐽 (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))
27 simp-4l 805 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝐽 ∈ (TopOn‘𝑋))
28 simprl 793 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑤𝐽)
291kqopn 21447 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽) → (𝐹𝑤) ∈ (KQ‘𝐽))
3027, 28, 29syl2anc 692 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → (𝐹𝑤) ∈ (KQ‘𝐽))
31 simprrl 803 . . . . . . . . . . . . . 14 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑧𝑤)
32 simplrl 799 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑧𝑋)
331kqfvima 21443 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤𝐽𝑧𝑋) → (𝑧𝑤 ↔ (𝐹𝑧) ∈ (𝐹𝑤)))
3427, 28, 32, 33syl3anc 1323 . . . . . . . . . . . . . 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 20641 . . . . . . . . . . . . . . . . . 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 4433 . . . . . . . . . . . . . . . . . 18 (𝑤𝐽𝑤 𝐽)
3938ad2antrl 763 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑤 𝐽)
40 eqid 2621 . . . . . . . . . . . . . . . . . 18 𝐽 = 𝐽
4140clscld 20761 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑤 𝐽) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽))
4237, 39, 41syl2anc 692 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽))
431kqcld 21448 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)))
4427, 42, 43syl2anc 692 . . . . . . . . . . . . . . 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 20770 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑤 𝐽) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤))
4637, 39, 45syl2anc 692 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤))
47 imass2 5460 . . . . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . 16 (KQ‘𝐽) = (KQ‘𝐽)
5049clsss2 20786 . . . . . . . . . . . . . . 15 (((𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤)))
5144, 48, 50syl2anc 692 . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → 𝐹 Fn 𝑋)
53 fnfun 5946 . . . . . . . . . . . . . . . 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 804 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧𝑋 ∧ (𝐹𝑧) ∈ 𝑎)) ∧ (𝑤𝐽 ∧ (𝑧𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)))) → ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎))
56 funimass2 5930 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝐹𝑎)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎)
5754, 55, 56syl2anc 692 . . . . . . . . . . . . . 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 3593 . . . . . . . . . . . . 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 2687 . . . . . . . . . . . . . . 15 (𝑚 = (𝐹𝑤) → ((𝐹𝑧) ∈ 𝑚 ↔ (𝐹𝑧) ∈ (𝐹𝑤)))
60 fveq2 6148 . . . . . . . . . . . . . . . 16 (𝑚 = (𝐹𝑤) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹𝑤)))
6160sseq1d 3611 . . . . . . . . . . . . . . 15 (𝑚 = (𝐹𝑤) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎 ↔ ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ 𝑎))
6259, 61anbi12d 746 . . . . . . . . . . . . . 14 (𝑚 = (𝐹𝑤) → (((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ ((𝐹𝑧) ∈ (𝐹𝑤) ∧ ((cls‘(KQ‘𝐽))‘(𝐹𝑤)) ⊆ 𝑎)))
6362rspcev 3295 . . . . . . . . . . . . 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 1321 . . . . . . . . . . . 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 3028 . . . . . . . . . . 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 2686 . . . . . . . . . . 11 ((𝐹𝑧) = 𝑏 → ((𝐹𝑧) ∈ 𝑎𝑏𝑎))
68 eleq1 2686 . . . . . . . . . . . . 13 ((𝐹𝑧) = 𝑏 → ((𝐹𝑧) ∈ 𝑚𝑏𝑚))
6968anbi1d 740 . . . . . . . . . . . 12 ((𝐹𝑧) = 𝑏 → (((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ (𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7069rexbidv 3045 . . . . . . . . . . 11 ((𝐹𝑧) = 𝑏 → (∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7167, 70imbi12d 334 . . . . . . . . . 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 445 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧𝑋) ∧ 𝑏𝑎) → ((𝐹𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7574an32s 845 . . . . . 6 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏𝑎) ∧ 𝑧𝑋) → ((𝐹𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
7675rexlimdva 3024 . . . . 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 678 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑎 ∈ (KQ‘𝐽) ∧ 𝑏𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
7978ralrimivva 2965 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ (KQ‘𝐽)∀𝑏𝑎𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))
80 isreg 21046 . 2 ((KQ‘𝐽) ∈ Reg ↔ ((KQ‘𝐽) ∈ Top ∧ ∀𝑎 ∈ (KQ‘𝐽)∀𝑏𝑎𝑚 ∈ (KQ‘𝐽)(𝑏𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))
815, 79, 80sylanbrc 697 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  wss 3555   cuni 4402  cmpt 4673  ccnv 5073  ran crn 5075  cima 5077  Fun wfun 5841   Fn wfn 5842  cfv 5847  (class class class)co 6604  Topctop 20617  TopOnctopon 20618  Clsdccld 20730  clsccl 20732   Cn ccn 20938  Regcreg 21023  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-int 4441  df-iun 4487  df-iin 4488  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-qtop 16088  df-top 20621  df-topon 20623  df-cld 20733  df-cls 20735  df-cn 20941  df-reg 21030  df-kq 21407
This theorem is referenced by:  kqreg  21464
  Copyright terms: Public domain W3C validator