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Theorem regr1lem 21744
Description: Lemma for regr1 21755. (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 472 . . . 4 ((𝜑𝐴𝑈) → 𝐽 ∈ Reg)
3 regr1lem.6 . . . . 5 (𝜑𝑈𝐽)
43adantr 472 . . . 4 ((𝜑𝐴𝑈) → 𝑈𝐽)
5 simpr 479 . . . 4 ((𝜑𝐴𝑈) → 𝐴𝑈)
6 regsep 21340 . . . 4 ((𝐽 ∈ Reg ∧ 𝑈𝐽𝐴𝑈) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))
72, 4, 5, 6syl3anc 1477 . . 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 764 . . . 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 768 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐽 ∈ (TopOn‘𝑋))
12 simplrl 819 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧𝐽)
13 kqval.2 . . . . . . . 8 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
1413kqopn 21739 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹𝑧) ∈ (KQ‘𝐽))
1511, 12, 14syl2anc 696 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹𝑧) ∈ (KQ‘𝐽))
16 toponuni 20921 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1711, 16syl 17 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑋 = 𝐽)
1817difeq1d 3870 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) = ( 𝐽 ∖ ((cls‘𝐽)‘𝑧)))
19 topontop 20920 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2011, 19syl 17 . . . . . . . . . 10 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐽 ∈ Top)
21 elssuni 4619 . . . . . . . . . . 11 (𝑧𝐽𝑧 𝐽)
2212, 21syl 17 . . . . . . . . . 10 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧 𝐽)
23 eqid 2760 . . . . . . . . . . 11 𝐽 = 𝐽
2423clscld 21053 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧 𝐽) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
2520, 22, 24syl2anc 696 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
2623cldopn 21037 . . . . . . . . 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 2839 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
2913kqopn 21739 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
3011, 28, 29syl2anc 696 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
31 simprrl 823 . . . . . . . 8 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐴𝑧)
3231adantr 472 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐴𝑧)
33 regr1lem.4 . . . . . . . . 9 (𝜑𝐴𝑋)
3433ad3antrrr 768 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐴𝑋)
3513kqfvima 21735 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽𝐴𝑋) → (𝐴𝑧 ↔ (𝐹𝐴) ∈ (𝐹𝑧)))
3611, 12, 34, 35syl3anc 1477 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐴𝑧 ↔ (𝐹𝐴) ∈ (𝐹𝑧)))
3732, 36mpbid 222 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹𝐴) ∈ (𝐹𝑧))
38 regr1lem.5 . . . . . . . . 9 (𝜑𝐵𝑋)
3938ad3antrrr 768 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐵𝑋)
40 simprrr 824 . . . . . . . . . 10 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → ((cls‘𝐽)‘𝑧) ⊆ 𝑈)
4140sseld 3743 . . . . . . . . 9 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → (𝐵 ∈ ((cls‘𝐽)‘𝑧) → 𝐵𝑈))
4241con3dimp 456 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ¬ 𝐵 ∈ ((cls‘𝐽)‘𝑧))
4339, 42eldifd 3726 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))
4413kqfvima 21735 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽𝐵𝑋) → (𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ↔ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
4511, 28, 39, 44syl3anc 1477 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ↔ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
4643, 45mpbid 222 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))))
4723sscls 21062 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧 𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
4820, 22, 47syl2anc 696 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
4948sscond 3890 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋𝑧))
50 imass2 5659 . . . . . . . 8 ((𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋𝑧) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ⊆ (𝐹 “ (𝑋𝑧)))
51 sslin 3982 . . . . . . . 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 21737 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅)
5411, 12, 53syl2anc 696 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅)
55 sseq0 4118 . . . . . . 7 ((((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) ⊆ ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) ∧ ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)
5652, 54, 55syl2anc 696 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)
57 eleq2 2828 . . . . . . . 8 (𝑚 = (𝐹𝑧) → ((𝐹𝐴) ∈ 𝑚 ↔ (𝐹𝐴) ∈ (𝐹𝑧)))
58 ineq1 3950 . . . . . . . . 9 (𝑚 = (𝐹𝑧) → (𝑚𝑛) = ((𝐹𝑧) ∩ 𝑛))
5958eqeq1d 2762 . . . . . . . 8 (𝑚 = (𝐹𝑧) → ((𝑚𝑛) = ∅ ↔ ((𝐹𝑧) ∩ 𝑛) = ∅))
6057, 593anbi13d 1550 . . . . . . 7 (𝑚 = (𝐹𝑧) → (((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝐴) ∈ (𝐹𝑧) ∧ (𝐹𝐵) ∈ 𝑛 ∧ ((𝐹𝑧) ∩ 𝑛) = ∅)))
61 eleq2 2828 . . . . . . . 8 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹𝐵) ∈ 𝑛 ↔ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
62 ineq2 3951 . . . . . . . . 9 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹𝑧) ∩ 𝑛) = ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
6362eqeq1d 2762 . . . . . . . 8 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → (((𝐹𝑧) ∩ 𝑛) = ∅ ↔ ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅))
6461, 633anbi23d 1551 . . . . . . 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 3463 . . . . . 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 1489 . . . . 5 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))
6766ex 449 . . . 4 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → (¬ 𝐵𝑈 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
689, 67mt3d 140 . . 3 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐵𝑈)
697, 68rexlimddv 3173 . 2 ((𝜑𝐴𝑈) → 𝐵𝑈)
7069ex 449 1 (𝜑 → (𝐴𝑈𝐵𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wrex 3051  {crab 3054  cdif 3712  cin 3714  wss 3715  c0 4058   cuni 4588  cmpt 4881  cima 5269  cfv 6049  Topctop 20900  TopOnctopon 20917  Clsdccld 21022  clsccl 21024  Regcreg 21315  KQckq 21698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-qtop 16369  df-top 20901  df-topon 20918  df-cld 21025  df-cls 21027  df-reg 21322  df-kq 21699
This theorem is referenced by:  regr1lem2  21745
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