cvmliftmolem2 - Mathbox for Mario Carneiro

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Theorem cvmliftmolem2 34828

Description: Lemma for cvmliftmo 34830. (Contributed by Mario Carneiro, 10-Mar-2015.)

**Hypotheses**
Ref	Expression
cvmliftmo.b	⊢ 𝐵 = ∪ 𝐶
cvmliftmo.y	⊢ 𝑌 = ∪ 𝐾
cvmliftmo.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftmo.k	⊢ (𝜑 → 𝐾 ∈ Conn)
cvmliftmo.l	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn)
cvmliftmo.o	⊢ (𝜑 → 𝑂 ∈ 𝑌)
cvmliftmoi.m	⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶))
cvmliftmoi.n	⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶))
cvmliftmoi.g	⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁))
cvmliftmoi.p	⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂))
cvmliftmolem.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})

**Assertion**
Ref	Expression
cvmliftmolem2	⊢ (𝜑 → 𝑀 = 𝑁)

Distinct variable groups: 𝑘,𝑠,𝑢,𝑣,𝐶 𝑘,𝐽,𝑠,𝑢,𝑣 𝑣,𝐵 𝐾,𝑠 𝑘,𝑀,𝑠,𝑢,𝑣 𝑁,𝑠 𝜑,𝑠 𝑘,𝐹,𝑠,𝑢,𝑣 𝑆,𝑠 𝑌,𝑠 Allowed substitution hints: 𝜑(𝑣,𝑢,𝑘) 𝐵(𝑢,𝑘,𝑠) 𝑆(𝑣,𝑢,𝑘) 𝐾(𝑣,𝑢,𝑘) 𝑁(𝑣,𝑢,𝑘) 𝑂(𝑣,𝑢,𝑘,𝑠) 𝑌(𝑣,𝑢,𝑘)

Proof of Theorem cvmliftmolem2 Dummy variables 𝑎 𝑏 𝑡 𝑦 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmliftmoi.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶))
2		cvmliftmo.y	. . . 4 ⊢ 𝑌 = ∪ 𝐾
3		cvmliftmo.b	. . . 4 ⊢ 𝐵 = ∪ 𝐶
4	2, 3	cnf 23137	. . 3 ⊢ (𝑀 ∈ (𝐾 Cn 𝐶) → 𝑀:𝑌⟶𝐵)
5		ffn 6716	. . 3 ⊢ (𝑀:𝑌⟶𝐵 → 𝑀 Fn 𝑌)
6	1, 4, 5	3syl 18	. 2 ⊢ (𝜑 → 𝑀 Fn 𝑌)
7		cvmliftmoi.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶))
8	2, 3	cnf 23137	. . 3 ⊢ (𝑁 ∈ (𝐾 Cn 𝐶) → 𝑁:𝑌⟶𝐵)
9		ffn 6716	. . 3 ⊢ (𝑁:𝑌⟶𝐵 → 𝑁 Fn 𝑌)
10	7, 8, 9	3syl 18	. 2 ⊢ (𝜑 → 𝑁 Fn 𝑌)
11		cvmliftmo.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Conn)
12		inss1 4224	. . . . . . 7 ⊢ (𝐾 ∩ (Clsd‘𝐾)) ⊆ 𝐾
13		cvmliftmo.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
14	13	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐹 ∈ (𝐶 CovMap 𝐽))
15	1, 4	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀:𝑌⟶𝐵)
16	15	ffvelcdmda 7088	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑀‘𝑥) ∈ 𝐵)
17		cvmcn 34808	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
18		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
19	3, 18	cnf 23137	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
20	13, 17, 19	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝐵⟶∪ 𝐽)
21	20	ffvelcdmda 7088	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀‘𝑥) ∈ 𝐵) → (𝐹‘(𝑀‘𝑥)) ∈ ∪ 𝐽)
22	16, 21	syldan 590	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(𝑀‘𝑥)) ∈ ∪ 𝐽)
23		cvmliftmolem.1	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})
24	23, 18	cvmcov 34809	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐹‘(𝑀‘𝑥)) ∈ ∪ 𝐽) → ∃𝑎 ∈ 𝐽 ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ (𝑆‘𝑎) ≠ ∅))
25	14, 22, 24	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ∃𝑎 ∈ 𝐽 ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ (𝑆‘𝑎) ≠ ∅))
26		n0 4342	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝑎) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ (𝑆‘𝑎))
27		cvmliftmo.l	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn)
28	27	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → 𝐾 ∈ 𝑛-Locally Conn)
29	1	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → 𝑀 ∈ (𝐾 Cn 𝐶))
30		simprrr 781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → 𝑡 ∈ (𝑆‘𝑎))
31	23	cvmsss 34813	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ (𝑆‘𝑎) → 𝑡 ⊆ 𝐶)
32	30, 31	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → 𝑡 ⊆ 𝐶)
33	13	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
34	15	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → 𝑀:𝑌⟶𝐵)
35		simprll 778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → 𝑥 ∈ 𝑌)
36	34, 35	ffvelcdmd 7089	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → (𝑀‘𝑥) ∈ 𝐵)
37		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → (𝐹‘(𝑀‘𝑥)) ∈ 𝑎)
38		eqid 2727	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏) = (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)
39	23, 3, 38	cvmsiota 34823	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑡 ∈ (𝑆‘𝑎) ∧ (𝑀‘𝑥) ∈ 𝐵 ∧ (𝐹‘(𝑀‘𝑥)) ∈ 𝑎)) → ((℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏) ∈ 𝑡 ∧ (𝑀‘𝑥) ∈ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)))
40	33, 30, 36, 37, 39	syl13anc 1370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → ((℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏) ∈ 𝑡 ∧ (𝑀‘𝑥) ∈ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)))
41	40	simpld 494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏) ∈ 𝑡)
42	32, 41	sseldd 3979	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏) ∈ 𝐶)
43		cnima 23156	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ (𝐾 Cn 𝐶) ∧ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏) ∈ 𝐶) → (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∈ 𝐾)
44	29, 42, 43	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∈ 𝐾)
45	40	simprd 495	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → (𝑀‘𝑥) ∈ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏))
46		elpreima 7061	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 Fn 𝑌 → (𝑥 ∈ (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ↔ (𝑥 ∈ 𝑌 ∧ (𝑀‘𝑥) ∈ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏))))
47	34, 5, 46	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → (𝑥 ∈ (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ↔ (𝑥 ∈ 𝑌 ∧ (𝑀‘𝑥) ∈ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏))))
48	35, 45, 47	mpbir2and 712	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → 𝑥 ∈ (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)))
49		nlly2i 23367	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ 𝑛-Locally Conn ∧ (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∈ 𝐾 ∧ 𝑥 ∈ (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏))) → ∃𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏))∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn))
50	28, 44, 48, 49	syl3anc 1369	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → ∃𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏))∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn))
51		simprr1 1219	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) ∧ ((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn))) → 𝑥 ∈ 𝑦)
52		cvmliftmo.o	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑂 ∈ 𝑌)
53		cvmliftmoi.g	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁))
54		cvmliftmoi.p	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂))
55		simplrr 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦)) → 𝑡 ∈ (𝑆‘𝑎))
56	55	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → 𝑡 ∈ (𝑆‘𝑎))
57	41	adantrr 716	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏) ∈ 𝑡)
58		simplll 774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦) → 𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)))
59	58	ad2antll 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → 𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)))
60	59	elpwid 4607	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → 𝑠 ⊆ (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)))
61		simplr3 1215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦) → (𝐾 ↾_t 𝑠) ∈ Conn)
62	61	ad2antll 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → (𝐾 ↾_t 𝑠) ∈ Conn)
63		simplr2 1214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦) → 𝑦 ⊆ 𝑠)
64	63	ad2antll 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → 𝑦 ⊆ 𝑠)
65		simprr1 1219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn))) → 𝑥 ∈ 𝑦)
66	65	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ ((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)))) → 𝑥 ∈ 𝑦)
67	66	adantrrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → 𝑥 ∈ 𝑦)
68	64, 67	sseldd 3979	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → 𝑥 ∈ 𝑠)
69		simprrr 781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → 𝑧 ∈ 𝑦)
70	64, 69	sseldd 3979	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → 𝑧 ∈ 𝑠)
71	37	adantrr 716	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → (𝐹‘(𝑀‘𝑥)) ∈ 𝑎)
72	3, 2, 13, 11, 27, 52, 1, 7, 53, 54, 23, 56, 57, 60, 62, 68, 68, 70, 71	cvmliftmolem1 34827	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → (𝑥 ∈ dom (𝑀 ∩ 𝑁) → 𝑧 ∈ dom (𝑀 ∩ 𝑁)))
73	3, 2, 13, 11, 27, 52, 1, 7, 53, 54, 23, 56, 57, 60, 62, 68, 70, 68, 71	cvmliftmolem1 34827	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → (𝑧 ∈ dom (𝑀 ∩ 𝑁) → 𝑥 ∈ dom (𝑀 ∩ 𝑁)))
74	72, 73	impbid 211	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦))) → (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))
75	74	anassrs 467	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) ∧ (((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn)) ∧ 𝑧 ∈ 𝑦)) → (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))
76	75	anassrs 467	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) ∧ ((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn))) ∧ 𝑧 ∈ 𝑦) → (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))
77	76	ralrimiva 3141	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) ∧ ((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn))) → ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))
78	51, 77	jca 511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) ∧ ((𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn))) → (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁))))
79	78	expr 456	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) ∧ (𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏)) ∧ 𝑦 ∈ 𝐾)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn) → (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
80	79	anassrs 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) ∧ 𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏))) ∧ 𝑦 ∈ 𝐾) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn) → (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
81	80	reximdva 3163	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) ∧ 𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏))) → (∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn) → ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
82	81	rexlimdva 3150	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → (∃𝑠 ∈ 𝒫 (^◡𝑀 “ (℩𝑏 ∈ 𝑡 (𝑀‘𝑥) ∈ 𝑏))∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑠 ∧ (𝐾 ↾_t 𝑠) ∈ Conn) → ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
83	50, 82	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎)))) → ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁))))
84	83	anassrs 467	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽)) ∧ ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ 𝑡 ∈ (𝑆‘𝑎))) → ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁))))
85	84	expr 456	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽)) ∧ (𝐹‘(𝑀‘𝑥)) ∈ 𝑎) → (𝑡 ∈ (𝑆‘𝑎) → ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
86	85	exlimdv 1929	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽)) ∧ (𝐹‘(𝑀‘𝑥)) ∈ 𝑎) → (∃𝑡 𝑡 ∈ (𝑆‘𝑎) → ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
87	26, 86	biimtrid 241	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽)) ∧ (𝐹‘(𝑀‘𝑥)) ∈ 𝑎) → ((𝑆‘𝑎) ≠ ∅ → ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
88	87	expimpd 453	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑎 ∈ 𝐽)) → (((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ (𝑆‘𝑎) ≠ ∅) → ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
89	88	anassrs 467	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑌) ∧ 𝑎 ∈ 𝐽) → (((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ (𝑆‘𝑎) ≠ ∅) → ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
90	89	rexlimdva 3150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (∃𝑎 ∈ 𝐽 ((𝐹‘(𝑀‘𝑥)) ∈ 𝑎 ∧ (𝑆‘𝑎) ≠ ∅) → ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
91	25, 90	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁))))
92	91	ralrimiva 3141	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁))))
93		conntop 23308	. . . . . . . . . 10 ⊢ (𝐾 ∈ Conn → 𝐾 ∈ Top)
94	11, 93	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
95		fndmin 7048	. . . . . . . . . . 11 ⊢ ((𝑀 Fn 𝑌 ∧ 𝑁 Fn 𝑌) → dom (𝑀 ∩ 𝑁) = {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)})
96	6, 10, 95	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑀 ∩ 𝑁) = {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)})
97		ssrab2 4073	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} ⊆ 𝑌
98	96, 97	eqsstrdi 4032	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑀 ∩ 𝑁) ⊆ 𝑌)
99	2	isclo 22978	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ dom (𝑀 ∩ 𝑁) ⊆ 𝑌) → (dom (𝑀 ∩ 𝑁) ∈ (𝐾 ∩ (Clsd‘𝐾)) ↔ ∀𝑥 ∈ 𝑌 ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
100	94, 98, 99	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (dom (𝑀 ∩ 𝑁) ∈ (𝐾 ∩ (Clsd‘𝐾)) ↔ ∀𝑥 ∈ 𝑌 ∃𝑦 ∈ 𝐾 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑧 ∈ dom (𝑀 ∩ 𝑁)))))
101	92, 100	mpbird 257	. . . . . . 7 ⊢ (𝜑 → dom (𝑀 ∩ 𝑁) ∈ (𝐾 ∩ (Clsd‘𝐾)))
102	12, 101	sselid 3976	. . . . . 6 ⊢ (𝜑 → dom (𝑀 ∩ 𝑁) ∈ 𝐾)
103		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑂 → (𝑀‘𝑥) = (𝑀‘𝑂))
104		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑂 → (𝑁‘𝑥) = (𝑁‘𝑂))
105	103, 104	eqeq12d 2743	. . . . . . . . . 10 ⊢ (𝑥 = 𝑂 → ((𝑀‘𝑥) = (𝑁‘𝑥) ↔ (𝑀‘𝑂) = (𝑁‘𝑂)))
106	105	elrab 3680	. . . . . . . . 9 ⊢ (𝑂 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} ↔ (𝑂 ∈ 𝑌 ∧ (𝑀‘𝑂) = (𝑁‘𝑂)))
107	52, 54, 106	sylanbrc 582	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)})
108	107, 96	eleqtrrd 2831	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ dom (𝑀 ∩ 𝑁))
109	108	ne0d 4331	. . . . . 6 ⊢ (𝜑 → dom (𝑀 ∩ 𝑁) ≠ ∅)
110		inss2 4225	. . . . . . 7 ⊢ (𝐾 ∩ (Clsd‘𝐾)) ⊆ (Clsd‘𝐾)
111	110, 101	sselid 3976	. . . . . 6 ⊢ (𝜑 → dom (𝑀 ∩ 𝑁) ∈ (Clsd‘𝐾))
112	2, 11, 102, 109, 111	connclo 23306	. . . . 5 ⊢ (𝜑 → dom (𝑀 ∩ 𝑁) = 𝑌)
113	112, 96	eqtr3d 2769	. . . 4 ⊢ (𝜑 → 𝑌 = {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)})
114		rabid2 3459	. . . 4 ⊢ (𝑌 = {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} ↔ ∀𝑥 ∈ 𝑌 (𝑀‘𝑥) = (𝑁‘𝑥))
115	113, 114	sylib 217	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 (𝑀‘𝑥) = (𝑁‘𝑥))
116	115	r19.21bi 3243	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑀‘𝑥) = (𝑁‘𝑥))
117	6, 10, 116	eqfnfvd 7037	1 ⊢ (𝜑 → 𝑀 = 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∃wrex 3065 {crab 3427 ∖ cdif 3941 ∩ cin 3943 ⊆ wss 3944 ∅c0 4318 𝒫 cpw 4598 {csn 4624 ∪ cuni 4903 ↦ cmpt 5225 ^◡ccnv 5671 dom cdm 5672 ↾ cres 5674 “ cima 5675 ∘ ccom 5676 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 ℩crio 7369 (class class class)co 7414 ↾_t crest 17393 Topctop 22782 Clsdccld 22907 Cn ccn 23115 Conncconn 23302 𝑛-Locally cnlly 23356 Homeochmeo 23644 CovMap ccvm 34801

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-map 8838 df-en 8956 df-fin 8959 df-fi 9426 df-rest 17395 df-topgen 17416 df-top 22783 df-topon 22800 df-bases 22836 df-cld 22910 df-nei 22989 df-cn 23118 df-conn 23303 df-nlly 23358 df-hmeo 23646 df-cvm 34802

This theorem is referenced by: cvmliftmoi 34829

W3C validator