MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  regr1lem Structured version   Visualization version   GIF version

Theorem regr1lem 21762
Description: Lemma for regr1 21773. (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 466 . . . 4 ((𝜑𝐴𝑈) → 𝐽 ∈ Reg)
3 regr1lem.6 . . . . 5 (𝜑𝑈𝐽)
43adantr 466 . . . 4 ((𝜑𝐴𝑈) → 𝑈𝐽)
5 simpr 471 . . . 4 ((𝜑𝐴𝑈) → 𝐴𝑈)
6 regsep 21358 . . . 4 ((𝐽 ∈ Reg ∧ 𝑈𝐽𝐴𝑈) → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))
72, 4, 5, 6syl3anc 1476 . . 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 705 . . . 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 709 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐽 ∈ (TopOn‘𝑋))
12 simplrl 762 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧𝐽)
13 kqval.2 . . . . . . . 8 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
1413kqopn 21757 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹𝑧) ∈ (KQ‘𝐽))
1511, 12, 14syl2anc 573 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹𝑧) ∈ (KQ‘𝐽))
16 toponuni 20938 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1711, 16syl 17 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑋 = 𝐽)
1817difeq1d 3878 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) = ( 𝐽 ∖ ((cls‘𝐽)‘𝑧)))
19 topontop 20937 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2011, 19syl 17 . . . . . . . . . 10 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐽 ∈ Top)
21 elssuni 4604 . . . . . . . . . . 11 (𝑧𝐽𝑧 𝐽)
2212, 21syl 17 . . . . . . . . . 10 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧 𝐽)
23 eqid 2771 . . . . . . . . . . 11 𝐽 = 𝐽
2423clscld 21071 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧 𝐽) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
2520, 22, 24syl2anc 573 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
2623cldopn 21055 . . . . . . . . 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 2850 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
2913kqopn 21757 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
3011, 28, 29syl2anc 573 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
31 simprrl 766 . . . . . . . 8 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐴𝑧)
3231adantr 466 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐴𝑧)
33 regr1lem.4 . . . . . . . . 9 (𝜑𝐴𝑋)
3433ad3antrrr 709 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐴𝑋)
3513kqfvima 21753 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽𝐴𝑋) → (𝐴𝑧 ↔ (𝐹𝐴) ∈ (𝐹𝑧)))
3611, 12, 34, 35syl3anc 1476 . . . . . . 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 709 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐵𝑋)
40 simprrr 767 . . . . . . . . . 10 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → ((cls‘𝐽)‘𝑧) ⊆ 𝑈)
4140sseld 3751 . . . . . . . . 9 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → (𝐵 ∈ ((cls‘𝐽)‘𝑧) → 𝐵𝑈))
4241con3dimp 395 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ¬ 𝐵 ∈ ((cls‘𝐽)‘𝑧))
4339, 42eldifd 3734 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))
4413kqfvima 21753 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽𝐵𝑋) → (𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ↔ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
4511, 28, 39, 44syl3anc 1476 . . . . . . 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 21080 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑧 𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
4820, 22, 47syl2anc 573 . . . . . . . . 9 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
4948sscond 3898 . . . . . . . 8 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋𝑧))
50 imass2 5641 . . . . . . . 8 ((𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋𝑧) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ⊆ (𝐹 “ (𝑋𝑧)))
51 sslin 3987 . . . . . . . 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 21755 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅)
5411, 12, 53syl2anc 573 . . . . . . 7 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅)
55 sseq0 4120 . . . . . . 7 ((((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) ⊆ ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) ∧ ((𝐹𝑧) ∩ (𝐹 “ (𝑋𝑧))) = ∅) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)
5652, 54, 55syl2anc 573 . . . . . 6 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)
57 eleq2 2839 . . . . . . . 8 (𝑚 = (𝐹𝑧) → ((𝐹𝐴) ∈ 𝑚 ↔ (𝐹𝐴) ∈ (𝐹𝑧)))
58 ineq1 3958 . . . . . . . . 9 (𝑚 = (𝐹𝑧) → (𝑚𝑛) = ((𝐹𝑧) ∩ 𝑛))
5958eqeq1d 2773 . . . . . . . 8 (𝑚 = (𝐹𝑧) → ((𝑚𝑛) = ∅ ↔ ((𝐹𝑧) ∩ 𝑛) = ∅))
6057, 593anbi13d 1549 . . . . . . 7 (𝑚 = (𝐹𝑧) → (((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅) ↔ ((𝐹𝐴) ∈ (𝐹𝑧) ∧ (𝐹𝐵) ∈ 𝑛 ∧ ((𝐹𝑧) ∩ 𝑛) = ∅)))
61 eleq2 2839 . . . . . . . 8 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹𝐵) ∈ 𝑛 ↔ (𝐹𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
62 ineq2 3959 . . . . . . . . 9 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹𝑧) ∩ 𝑛) = ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
6362eqeq1d 2773 . . . . . . . 8 (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → (((𝐹𝑧) ∩ 𝑛) = ∅ ↔ ((𝐹𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅))
6461, 633anbi23d 1550 . . . . . . 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 3474 . . . . . 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 1488 . . . . 5 ((((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵𝑈) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))
6766ex 397 . . . 4 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → (¬ 𝐵𝑈 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅)))
689, 67mt3d 142 . . 3 (((𝜑𝐴𝑈) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐵𝑈)
697, 68rexlimddv 3183 . 2 ((𝜑𝐴𝑈) → 𝐵𝑈)
7069ex 397 1 (𝜑 → (𝐴𝑈𝐵𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wrex 3062  {crab 3065  cdif 3720  cin 3722  wss 3723  c0 4063   cuni 4575  cmpt 4864  cima 5253  cfv 6030  Topctop 20917  TopOnctopon 20934  Clsdccld 21040  clsccl 21042  Regcreg 21333  KQckq 21716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-qtop 16374  df-top 20918  df-topon 20935  df-cld 21043  df-cls 21045  df-reg 21340  df-kq 21717
This theorem is referenced by:  regr1lem2  21763
  Copyright terms: Public domain W3C validator