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Theorem regr1lem 23729
Description: Lemma for regr1 23740. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypotheses
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
regr1lem.2 (𝜑𝐽 ∈ (TopOn‘𝑋))
regr1lem.3 (𝜑𝐽 ∈ Reg)
regr1lem.4 (𝜑𝐴𝑋)
regr1lem.5 (𝜑𝐵𝑋)
regr1lem.6 (𝜑𝑈𝐽)
regr1lem.7 (𝜑 → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))
Assertion
Ref Expression
regr1lem (𝜑 → (𝐴𝑈𝐵𝑈))
Distinct variable groups:   𝑚,𝑛,𝑥,𝑦,𝐴   𝐵,𝑚,𝑛,𝑥,𝑦   𝑚,𝐽,𝑛,𝑥,𝑦   𝑚,𝐹,𝑛   𝑚,𝑋,𝑛,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚,𝑛)   𝑈(𝑥,𝑦,𝑚,𝑛)   𝐹(𝑥,𝑦)

Proof of Theorem regr1lem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 regr1lem.3 . . . . 5 (𝜑𝐽 ∈ Reg)
21adantr 479 . . . 4 ((𝜑𝐴𝑈) → 𝐽 ∈ Reg)
3 regr1lem.6 . . . . 5 (𝜑𝑈𝐽)
43adantr 479 . . . 4 ((𝜑𝐴𝑈) → 𝑈𝐽)
5 simpr 483 . . . 4 ((𝜑𝐴𝑈) → 𝐴𝑈)
6 regsep 23324 . . . 4 ((𝐽 ∈ Reg ∧ 𝑈𝐽𝐴𝑈) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))
72, 4, 5, 6syl3anc 1368 . . 3 ((𝜑𝐴𝑈) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))
8 regr1lem.7 . . . . 5 (𝜑 → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))
98ad2antrr 724 . . . 4 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))
10 regr1lem.2 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘𝑋))
1110ad3antrrr 728 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐽 ∈ (TopOn‘𝑋))
12 simplrl 775 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧𝐽)
13 kqval.2 . . . . . . . 8 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
1413kqopn 23724 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹𝑧) ∈ (KQ‘𝐽))
1511, 12, 14syl2anc 582 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹𝑧) ∈ (KQ‘𝐽))
16 toponuni 22902 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1711, 16syl 17 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑋 = 𝐽)
1817difeq1d 4118 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) = ( 𝐽 ∖ ((cls‘𝐽)‘𝑧)))
19 topontop 22901 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2011, 19syl 17 . . . . . . . . . 10 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐽 ∈ Top)
21 elssuni 4938 . . . . . . . . . . 11 (𝑧𝐽𝑧 𝐽)
2212, 21syl 17 . . . . . . . . . 10 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧 𝐽)
23 eqid 2726 . . . . . . . . . . 11 𝐽 = 𝐽
2423clscld 23037 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧 𝐽) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
2520, 22, 24syl2anc 582 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
2623cldopn 23021 . . . . . . . . 9 (((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽) → ( 𝐽 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
2725, 26syl 17 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ( 𝐽 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
2818, 27eqeltrd 2826 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
2913kqopn 23724 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
3011, 28, 29syl2anc 582 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
31 simprrl 779 . . . . . . . 8 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐴𝑧)
3231adantr 479 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐴𝑧)
33 regr1lem.4 . . . . . . . . 9 (𝜑𝐴𝑋)
3433ad3antrrr 728 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐴𝑋)
3513kqfvima 23720 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽𝐴𝑋) → (𝐴𝑧 ↔ (𝐹𝐴) ∈ (𝐹𝑧)))
3611, 12, 34, 35syl3anc 1368 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐴𝑧 ↔ (𝐹𝐴) ∈ (𝐹𝑧)))
3732, 36mpbid 231 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹𝐴) ∈ (𝐹𝑧))
38 regr1lem.5 . . . . . . . . 9 (𝜑𝐵𝑋)
3938ad3antrrr 728 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐵𝑋)
40 simprrr 780 . . . . . . . . . 10 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → ((cls‘𝐽)‘𝑧) ⊆ 𝑈)
4140sseld 3978 . . . . . . . . 9 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → (𝐵 ∈ ((cls‘𝐽)‘𝑧) → 𝐵𝑈))
4241con3dimp 407 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ¬ 𝐵 ∈ ((cls‘𝐽)‘𝑧))
4339, 42eldifd 3958 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))
4413kqfvima 23720 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽𝐵𝑋) → (𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ↔ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
4511, 28, 39, 44syl3anc 1368 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ↔ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
4643, 45mpbid 231 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))))
4723sscls 23046 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧 𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
4820, 22, 47syl2anc 582 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
4948sscond 4139 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋𝑧))
50 imass2 6103 . . . . . . . 8 ((𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋𝑧) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ⊆ (𝐹 “ (𝑋𝑧)))
51 sslin 4234 . . . . . . . 8 ((𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ⊆ (𝐹 “ (𝑋𝑧)) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) ⊆ ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))))
5249, 50, 513syl 18 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) ⊆ ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))))
5313kqdisj 23722 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅)
5411, 12, 53syl2anc 582 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅)
55 sseq0 4398 . . . . . . 7 ((((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) ⊆ ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) ∧ ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)
5652, 54, 55syl2anc 582 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)
57 eleq2 2815 . . . . . . . 8 (𝑚 = (𝐹𝑧) → ((𝐹𝐴) ∈ 𝑚 ↔ (𝐹𝐴) ∈ (𝐹𝑧)))
58 ineq1 4204 . . . . . . . . 9 (𝑚 = (𝐹𝑧) → (𝑚𝑛) = ((𝐹𝑧) ∩ 𝑛))
5958eqeq1d 2728 . . . . . . . 8 (𝑚 = (𝐹𝑧) → ((𝑚𝑛) = ∅ ↔ ((𝐹𝑧) ∩ 𝑛) = ∅))
6057, 593anbi13d 1435 . . . . . . 7 (𝑚 = (𝐹𝑧) → (((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝐴) ∈ (𝐹𝑧) ∧ (𝐹𝐵) ∈ 𝑛 ∧ ((𝐹𝑧) ∩ 𝑛) = ∅)))
61 eleq2 2815 . . . . . . . 8 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹𝐵) ∈ 𝑛 ↔ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
62 ineq2 4205 . . . . . . . . 9 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹𝑧) ∩ 𝑛) = ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
6362eqeq1d 2728 . . . . . . . 8 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → (((𝐹𝑧) ∩ 𝑛) = ∅ ↔ ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅))
6461, 633anbi23d 1436 . . . . . . 7 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → (((𝐹𝐴) ∈ (𝐹𝑧) ∧ (𝐹𝐵) ∈ 𝑛 ∧ ((𝐹𝑧) ∩ 𝑛) = ∅) ↔ ((𝐹𝐴) ∈ (𝐹𝑧) ∧ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∧ ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)))
6560, 64rspc2ev 3621 . . . . . 6 (((𝐹𝑧) ∈ (KQ‘𝐽) ∧ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽) ∧ ((𝐹𝐴) ∈ (𝐹𝑧) ∧ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∧ ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))
6615, 30, 37, 46, 56, 65syl113anc 1379 . . . . 5 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))
6766ex 411 . . . 4 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → (¬ 𝐵𝑈 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
689, 67mt3d 148 . . 3 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐵𝑈)
697, 68rexlimddv 3151 . 2 ((𝜑𝐴𝑈) → 𝐵𝑈)
7069ex 411 1 (𝜑 → (𝐴𝑈𝐵𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wrex 3060  {crab 3420  cdif 3944  cin 3946  wss 3947  c0 4323   cuni 4906  cmpt 5227  cima 5676  cfv 6544  Topctop 22881  TopOnctopon 22898  Clsdccld 23006  clsccl 23008  Regcreg 23299  KQckq 23683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-iin 4997  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7417  df-oprab 7418  df-mpo 7419  df-qtop 17515  df-top 22882  df-topon 22899  df-cld 23009  df-cls 23011  df-reg 23306  df-kq 23684
This theorem is referenced by:  regr1lem2  23730
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