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Theorem regr1lem 21352
Description: Lemma for regr1 21363. (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 480 . . . 4 ((𝜑𝐴𝑈) → 𝐽 ∈ Reg)
3 regr1lem.6 . . . . 5 (𝜑𝑈𝐽)
43adantr 480 . . . 4 ((𝜑𝐴𝑈) → 𝑈𝐽)
5 simpr 476 . . . 4 ((𝜑𝐴𝑈) → 𝐴𝑈)
6 regsep 20948 . . . 4 ((𝐽 ∈ Reg ∧ 𝑈𝐽𝐴𝑈) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))
72, 4, 5, 6syl3anc 1318 . . 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 758 . . . 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 762 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐽 ∈ (TopOn‘𝑋))
12 simplrl 796 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧𝐽)
13 kqval.2 . . . . . . . 8 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
1413kqopn 21347 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹𝑧) ∈ (KQ‘𝐽))
1511, 12, 14syl2anc 691 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹𝑧) ∈ (KQ‘𝐽))
16 toponuni 20542 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1711, 16syl 17 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑋 = 𝐽)
1817difeq1d 3689 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) = ( 𝐽 ∖ ((cls‘𝐽)‘𝑧)))
19 topontop 20541 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2011, 19syl 17 . . . . . . . . . 10 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐽 ∈ Top)
21 elssuni 4403 . . . . . . . . . . 11 (𝑧𝐽𝑧 𝐽)
2212, 21syl 17 . . . . . . . . . 10 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧 𝐽)
23 eqid 2610 . . . . . . . . . . 11 𝐽 = 𝐽
2423clscld 20661 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧 𝐽) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
2520, 22, 24syl2anc 691 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
2623cldopn 20645 . . . . . . . . 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 2688 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
2913kqopn 21347 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
3011, 28, 29syl2anc 691 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
31 simprrl 800 . . . . . . . 8 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐴𝑧)
3231adantr 480 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐴𝑧)
33 regr1lem.4 . . . . . . . . 9 (𝜑𝐴𝑋)
3433ad3antrrr 762 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐴𝑋)
3513kqfvima 21343 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽𝐴𝑋) → (𝐴𝑧 ↔ (𝐹𝐴) ∈ (𝐹𝑧)))
3611, 12, 34, 35syl3anc 1318 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐴𝑧 ↔ (𝐹𝐴) ∈ (𝐹𝑧)))
3732, 36mpbid 221 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹𝐴) ∈ (𝐹𝑧))
38 regr1lem.5 . . . . . . . . 9 (𝜑𝐵𝑋)
3938ad3antrrr 762 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐵𝑋)
40 simprrr 801 . . . . . . . . . 10 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → ((cls‘𝐽)‘𝑧) ⊆ 𝑈)
4140sseld 3567 . . . . . . . . 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 3551 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))
4413kqfvima 21343 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽𝐵𝑋) → (𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ↔ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
4511, 28, 39, 44syl3anc 1318 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ↔ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
4643, 45mpbid 221 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))))
4723sscls 20670 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧 𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
4820, 22, 47syl2anc 691 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
4948sscond 3709 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋𝑧))
50 imass2 5420 . . . . . . . 8 ((𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋𝑧) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ⊆ (𝐹 “ (𝑋𝑧)))
51 sslin 3801 . . . . . . . 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 21345 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅)
5411, 12, 53syl2anc 691 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅)
55 sseq0 3927 . . . . . . 7 ((((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) ⊆ ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) ∧ ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)
5652, 54, 55syl2anc 691 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)
57 eleq2 2677 . . . . . . . 8 (𝑚 = (𝐹𝑧) → ((𝐹𝐴) ∈ 𝑚 ↔ (𝐹𝐴) ∈ (𝐹𝑧)))
58 ineq1 3769 . . . . . . . . 9 (𝑚 = (𝐹𝑧) → (𝑚𝑛) = ((𝐹𝑧) ∩ 𝑛))
5958eqeq1d 2612 . . . . . . . 8 (𝑚 = (𝐹𝑧) → ((𝑚𝑛) = ∅ ↔ ((𝐹𝑧) ∩ 𝑛) = ∅))
6057, 593anbi13d 1393 . . . . . . 7 (𝑚 = (𝐹𝑧) → (((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝐴) ∈ (𝐹𝑧) ∧ (𝐹𝐵) ∈ 𝑛 ∧ ((𝐹𝑧) ∩ 𝑛) = ∅)))
61 eleq2 2677 . . . . . . . 8 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹𝐵) ∈ 𝑛 ↔ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
62 ineq2 3770 . . . . . . . . 9 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹𝑧) ∩ 𝑛) = ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
6362eqeq1d 2612 . . . . . . . 8 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → (((𝐹𝑧) ∩ 𝑛) = ∅ ↔ ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅))
6461, 633anbi23d 1394 . . . . . . 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 3295 . . . . . 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 1330 . . . . 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 139 . . 3 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐵𝑈)
697, 68rexlimddv 3017 . 2 ((𝜑𝐴𝑈) → 𝐵𝑈)
7069ex 449 1 (𝜑 → (𝐴𝑈𝐵𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wrex 2897  {crab 2900  cdif 3537  cin 3539  wss 3540  c0 3874   cuni 4372  cmpt 4643  cima 5041  cfv 5804  Topctop 20517  TopOnctopon 20518  Clsdccld 20630  clsccl 20632  Regcreg 20923  KQckq 21306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-qtop 15990  df-top 20521  df-topon 20523  df-cld 20633  df-cls 20635  df-reg 20930  df-kq 21307
This theorem is referenced by:  regr1lem2  21353
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