Theorem regr1lem 23763

Description: Lemma for regr1 23774. (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 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝐽 ∈ Reg)
3		regr1lem.6	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝑈 ∈ 𝐽)
5		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝑈)
6		regsep 23358	. . . 4 ⊢ ((𝐽 ∈ Reg ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ∃𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))
7	2, 4, 5, 6	syl3anc 1370	. . 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 726	. . . 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 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝐽 ∈ (TopOn‘𝑋))
12		simplrl 777	. . . . . . 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 23758	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
15	11, 12, 14	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
16		toponuni 22936	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
17	11, 16	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝑋 = ∪ 𝐽)
18	17	difeq1d 4135	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) = (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑧)))
19		topontop 22935	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
20	11, 19	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝐽 ∈ Top)
21		elssuni 4942	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽)
22	12, 21	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝑧 ⊆ ∪ 𝐽)
23		eqid 2735	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	clscld 23071	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
25	20, 22, 24	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → ((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽))
26	23	cldopn 23055	. . . . . . . . 9 ⊢ (((cls‘𝐽)‘𝑧) ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
27	25, 26	syl 17	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (∪ 𝐽 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
28	18, 27	eqeltrd 2839	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽)
29	13	kqopn 23758	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ∈ 𝐽) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
30	11, 28, 29	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ∈ (KQ‘𝐽))
31		simprrl 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐴 ∈ 𝑧)
32	31	adantr 480	. . . . . . 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 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝐴 ∈ 𝑋)
35	13	kqfvima 23754	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑧 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑧)))
36	11, 12, 34, 35	syl3anc 1370	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝐴 ∈ 𝑧 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑧)))
37	32, 36	mpbid 232	. . . . . 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 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑋)
40		simprrr 782	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → ((cls‘𝐽)‘𝑧) ⊆ 𝑈)
41	40	sseld 3994	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → (𝐵 ∈ ((cls‘𝐽)‘𝑧) → 𝐵 ∈ 𝑈))
42	41	con3dimp 408	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → ¬ 𝐵 ∈ ((cls‘𝐽)‘𝑧))
43	39, 42	eldifd 3974	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝐵 ∈ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))
44	13	kqfvima 23754	. . . . . . . 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 1370	. . . . . . 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 232	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝐹‘𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))))
47	23	sscls 23080	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ ∪ 𝐽) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
48	20, 22, 47	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → 𝑧 ⊆ ((cls‘𝐽)‘𝑧))
49	48	sscond 4156	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → (𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋 ∖ 𝑧))
50		imass2 6123	. . . . . . . 8 ⊢ ((𝑋 ∖ ((cls‘𝐽)‘𝑧)) ⊆ (𝑋 ∖ 𝑧) → (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) ⊆ (𝐹 “ (𝑋 ∖ 𝑧)))
51		sslin 4251	. . . . . . . 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 18	. . . . . . 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 23756	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ 𝑧))) = ∅)
54	11, 12, 53	syl2anc 584	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ 𝑧))) = ∅)
55		sseq0 4409	. . . . . . 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 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) ∧ ¬ 𝐵 ∈ 𝑈) → ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅)
57		eleq2 2828	. . . . . . . 8 ⊢ (𝑚 = (𝐹 “ 𝑧) → ((𝐹‘𝐴) ∈ 𝑚 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑧)))
58		ineq1 4221	. . . . . . . . 9 ⊢ (𝑚 = (𝐹 “ 𝑧) → (𝑚 ∩ 𝑛) = ((𝐹 “ 𝑧) ∩ 𝑛))
59	58	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑚 = (𝐹 “ 𝑧) → ((𝑚 ∩ 𝑛) = ∅ ↔ ((𝐹 “ 𝑧) ∩ 𝑛) = ∅))
60	57, 59	3anbi13d 1437	. . . . . . 7 ⊢ (𝑚 = (𝐹 “ 𝑧) → (((𝐹‘𝐴) ∈ 𝑚 ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ((𝐹‘𝐴) ∈ (𝐹 “ 𝑧) ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ ((𝐹 “ 𝑧) ∩ 𝑛) = ∅)))
61		eleq2 2828	. . . . . . . 8 ⊢ (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹‘𝐵) ∈ 𝑛 ↔ (𝐹‘𝐵) ∈ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
62		ineq2 4222	. . . . . . . . 9 ⊢ (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → ((𝐹 “ 𝑧) ∩ 𝑛) = ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))))
63	62	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑛 = (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧))) → (((𝐹 “ 𝑧) ∩ 𝑛) = ∅ ↔ ((𝐹 “ 𝑧) ∩ (𝐹 “ (𝑋 ∖ ((cls‘𝐽)‘𝑧)))) = ∅))
64	61, 63	3anbi23d 1438	. . . . . . 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 3635	. . . . . 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 1381	. . . . 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 412	. . . 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 148	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑈))) → 𝐵 ∈ 𝑈)
69	7, 68	rexlimddv 3159	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝐵 ∈ 𝑈)
70	69	ex 412	1 ⊢ (𝜑 → (𝐴 ∈ 𝑈 → 𝐵 ∈ 𝑈))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  {crab 3433   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  ∪ cuni 4912   ↦ cmpt 5231   “ cima 5692  ‘cfv 6563  Topctop 22915  TopOnctopon 22932  Clsdccld 23040  clsccl 23042  Regcreg 23333  KQckq 23717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17554  df-top 22916  df-topon 22933  df-cld 23043  df-cls 23045  df-reg 23340  df-kq 23718

This theorem is referenced by:  regr1lem2  23764