Theorem regr1lem 24113

Description: Lemma for regr1 24124. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypotheses

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
regr1lem.2	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
regr1lem.3	⊢ (𝜑 → 𝐽 ∈ Reg)
regr1lem.4	⊢ (𝜑 → 𝐴 ∈ 𝑋)
regr1lem.5	⊢ (𝜑 → 𝐵 ∈ 𝑋)
regr1lem.6	⊢ (𝜑 → 𝑈 ∈ 𝐽)
regr1lem.7	⊢ (𝜑 → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝐴) ∈ 𝑚 ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))

Assertion

Ref	Expression
regr1lem	⊢ (𝜑 → (𝐴 ∈ 𝑈 → 𝐵 ∈ 𝑈))

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.

Step	Hyp	Ref	Expression
1		regr1lem.3	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Reg)
2	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝐽 ∈ Reg)
3		regr1lem.6	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
4	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝑈 ∈ 𝐽)
5		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝑈)
6		regsep 23707	. . . 4 ⊢ ((𝐽 ∈ Reg ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ∃𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))
7	2, 4, 5, 6	syl3anc 1399	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ∃𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))
8		regr1lem.7	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝐴) ∈ 𝑚 ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
9	8	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝐴) ∈ 𝑚 ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
10		regr1lem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
11	10	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝐽 ∈ (TopOn‘𝑋))
12		simplrl 789	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝑧 ∈ 𝐽)
13		kqval.2	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
14	13	kqopn 24108	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
15	11, 12, 14	syl2anc 596	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
16		toponuni 23287	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
17	11, 16	syl 18	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝑋 = ∪ 𝐽)
18	17	difeq1d 4065	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑧)))
19		topontop 23286	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
20	11, 19	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝐽 ∈ Top)
21		elssuni 4904	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
22	12, 21	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝑧 ⊆ ∪ 𝐽)
23		eqid 2763	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	clscld 23420	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
25	20, 22, 24	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
26	23	cldopn 23404	. . . . . . . . 9 ⊢ (((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
27	25, 26	syl 18	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
28	18, 27	eqeltrd 2864	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
29	13	kqopn 24108	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
30	11, 28, 29	syl2anc 596	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
31		simprrl 793	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐴 ∈ 𝑧)
32	31	adantr 486	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝐴 ∈ 𝑧)
33		regr1lem.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
34	33	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝐴 ∈ 𝑋)
35	13	kqfvima 24104	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑧 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑧)))
36	11, 12, 34, 35	syl3anc 1399	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝐴 ∈ 𝑧 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑧)))
37	32, 36	mpbid 235	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝐹‘𝐴) ∈ (𝐹 “ 𝑧))
38		regr1lem.5	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑋)
39	38	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑋)
40		simprrr 794	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → ((cls‘𝐽)‘𝑧) ⊆ 𝑈)
41	40	sseld 3912	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → (𝐵 ∈ ((cls‘𝐽)‘𝑧) → 𝐵 ∈ 𝑈))
42	41	con3dimp 414	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → ¬ 𝐵 ∈ ((cls‘𝐽)‘𝑧))
43	39, 42	eldifd 4053	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))
44	13	kqfvima 24104	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽 ∧ 𝐵 ∈ 𝑋) → (𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ↔ (𝐹‘𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
45	11, 28, 39, 44	syl3anc 1399	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ↔ (𝐹‘𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
46	43, 45	mpbid 235	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝐹‘𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))))
47	23	sscls 23429	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ ∪ 𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
48	20, 22, 47	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
49	48	sscond 4085	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋 ∖ 𝑧))
50		imass2 6073	. . . . . . . 8 ⊢ ((𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋 ∖ 𝑧) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ⊆ (𝐹 “ (𝑋 ∖ 𝑧)))
51		sslin 4142	. . . . . . . 8 ⊢ ((𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ⊆ (𝐹 “ (𝑋 ∖ 𝑧)) → ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) ⊆ ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ 𝑧))))
52	49, 50, 51	3syl 19	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) ⊆ ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ 𝑧))))
53	13	kqdisj 24106	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ 𝑧))) = ∅)
54	11, 12, 53	syl2anc 596	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ 𝑧))) = ∅)
55		sseq0 4357	. . . . . . 7 ⊢ ((((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) ⊆ ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ 𝑧))) ∧ ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ 𝑧))) = ∅) → ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)
56	52, 54, 55	syl2anc 596	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)
57		eleq2 2853	. . . . . . . 8 ⊢ (𝑚 = (𝐹 “ 𝑧) → ((𝐹‘𝐴) ∈ 𝑚 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑧)))
58		ineq1 4113	. . . . . . . . 9 ⊢ (𝑚 = (𝐹 “ 𝑧) → (𝑚 ∩ 𝑛) = ((𝐹 “ 𝑧) ∩ 𝑛))
59	58	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑚 = (𝐹 “ 𝑧) → ((𝑚 ∩ 𝑛) = ∅ ↔ ((𝐹 “ 𝑧) ∩ 𝑛) = ∅))
60	57, 59	3anbi13d 1467	. . . . . . 7 ⊢ (𝑚 = (𝐹 “ 𝑧) → (((𝐹‘𝐴) ∈ 𝑚 ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝐴) ∈ (𝐹 “ 𝑧) ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ ((𝐹 “ 𝑧) ∩ 𝑛) = ∅)))
61		eleq2 2853	. . . . . . . 8 ⊢ (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹‘𝐵) ∈ 𝑛 ↔ (𝐹‘𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
62		ineq2 4114	. . . . . . . . 9 ⊢ (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹 “ 𝑧) ∩ 𝑛) = ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
63	62	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → (((𝐹 “ 𝑧) ∩ 𝑛) = ∅ ↔ ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅))
64	61, 63	3anbi23d 1468	. . . . . . 7 ⊢ (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → (((𝐹‘𝐴) ∈ (𝐹 “ 𝑧) ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ ((𝐹 “ 𝑧) ∩ 𝑛) = ∅) ↔ ((𝐹‘𝐴) ∈ (𝐹 “ 𝑧) ∧ (𝐹‘𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∧ ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)))
65	60, 64	rspc2ev 3592	. . . . . 6 ⊢ (((𝐹 “ 𝑧) ∈ (KQ‘𝐽) ∧ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽) ∧ ((𝐹‘𝐴) ∈ (𝐹 “ 𝑧) ∧ (𝐹‘𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∧ ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝐴) ∈ 𝑚 ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
66	15, 30, 37, 46, 56, 65	syl113anc 1410	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝐴) ∈ 𝑚 ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
67	66	ex 418	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → (¬ 𝐵 ∈ 𝑈 → ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝐴) ∈ 𝑚 ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
68	9, 67	mt3d 149	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐵 ∈ 𝑈)
69	7, 68	rexlimddv 3173	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝐵 ∈ 𝑈)
70	69	ex 418	1 ⊢ (𝜑 → (𝐴 ∈ 𝑈 → 𝐵 ∈ 𝑈))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  ∅c0 4281  ∪ cuni 4872   ↦ cmpt 5192   “ cima 5661  ‘cfv 6566  Topctop 23266  TopOnctopon 23283  Clsdccld 23389  clsccl 23391  Regcreg 23682  KQckq 24067

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-qtop 17729  df-top 23267  df-topon 23284  df-cld 23392  df-cls 23394  df-reg 23689  df-kq 24068

This theorem is used by:  regr1lem2  24114