cvmopnlem - Mathbox for Mario Carneiro

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Theorem cvmopnlem 35791

Description: Lemma for cvmopn 35793. (Contributed by Mario Carneiro, 7-May-2015.)

**Hypotheses**
Ref	Expression
cvmcov.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))))})
cvmseu.1	⊢ 𝐵 = ∪ 𝐶

**Assertion**
Ref	Expression
cvmopnlem	⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → (𝐹 “ 𝐴) ∈ 𝐽)

Distinct variable groups: 𝑘,𝑠,𝑢,𝑣,𝐶 𝑘,𝐹,𝑠,𝑢,𝑣 𝑘,𝐽,𝑠,𝑢,𝑣 𝑢,𝐴,𝑣 𝑣,𝐵 Allowed substitution hints: 𝐴(𝑘, 𝑠) 𝐵(𝑢, 𝑘, 𝑠) 𝑆(𝑣, 𝑢, 𝑘, 𝑠)

Proof of Theorem cvmopnlem Dummy variables 𝑡 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 779	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → 𝐹 ∈ (𝐶 CovMap 𝐽))
2		cvmcn 35775	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
3	2	adantr 486	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → 𝐹 ∈ (𝐶 Cn 𝐽))
4		cvmseu.1	. . . . . . . . . 10 ⊢ 𝐵 = ∪ 𝐶
5		eqid 2766	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
6	4, 5	cnf 23440	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
7	3, 6	syl 18	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → 𝐹:𝐵⟶∪ 𝐽)
8	7	adantr 486	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → 𝐹:𝐵⟶∪ 𝐽)
9		elssuni 4909	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ⊆ ∪ 𝐶)
10	9, 4	sseqtrrdi 3981	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ⊆ 𝐵)
11	10	adantl 487	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → 𝐴 ⊆ 𝐵)
12	11	sselda 3940	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐵)
13	8, 12	ffvelcdmd 7087	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ∪ 𝐽)
14		cvmcov.1	. . . . . . 7 ⊢ 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))))})
15	14, 5	cvmcov 35776	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐹‘𝑧) ∈ ∪ 𝐽) → ∃𝑡 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑡 ∧ (𝑆‘𝑡) ≠ ∅))
16	1, 13, 15	syl2anc 596	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → ∃𝑡 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑡 ∧ (𝑆‘𝑡) ≠ ∅))
17		n0 4310	. . . . . . . 8 ⊢ ((𝑆‘𝑡) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑆‘𝑡))
18		inss2 4193	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ⊆ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)
19		resima2 6020	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ⊆ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥) → ((𝐹 ↾ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) = (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))))
20	18, 19	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) = (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)))
21		simprr 785	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝑤 ∈ (𝑆‘𝑡))
22	1	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
23	12	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝑧 ∈ 𝐵)
24		simprl 783	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → (𝐹‘𝑧) ∈ 𝑡)
25		eqid 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥) = (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)
26	14, 4, 25	cvmsiota 35790	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑤 ∈ (𝑆‘𝑡) ∧ 𝑧 ∈ 𝐵 ∧ (𝐹‘𝑧) ∈ 𝑡)) → ((℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥) ∈ 𝑤 ∧ 𝑧 ∈ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)))
27	22, 21, 23, 24, 26	syl13anc 1399	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → ((℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥) ∈ 𝑤 ∧ 𝑧 ∈ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)))
28	27	simpld 500	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥) ∈ 𝑤)
29	14	cvmshmeo 35784	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ (𝑆‘𝑡) ∧ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥) ∈ 𝑤) → (𝐹 ↾ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ∈ ((𝐶 ↾_t (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))Homeo(𝐽 ↾_t 𝑡)))
30	21, 28, 29	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → (𝐹 ↾ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ∈ ((𝐶 ↾_t (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))Homeo(𝐽 ↾_t 𝑡)))
31		cvmtop1 35773	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
32	22, 31	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝐶 ∈ Top)
33		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝐴 ∈ 𝐶)
34		elrestr 17506	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ Top ∧ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥) ∈ 𝑤 ∧ 𝐴 ∈ 𝐶) → (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ∈ (𝐶 ↾_t (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)))
35	32, 28, 33, 34	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ∈ (𝐶 ↾_t (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)))
36		hmeoima 23959	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ↾ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ∈ ((𝐶 ↾_t (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))Homeo(𝐽 ↾_t 𝑡)) ∧ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ∈ (𝐶 ↾_t (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) → ((𝐹 ↾ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∈ (𝐽 ↾_t 𝑡))
37	30, 35, 36	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → ((𝐹 ↾ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∈ (𝐽 ↾_t 𝑡))
38	20, 37	eqeltrrid 2871	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∈ (𝐽 ↾_t 𝑡))
39		cvmtop2 35774	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
40	39	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → 𝐽 ∈ Top)
41	40	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝐽 ∈ Top)
42	14	cvmsrcl 35777	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (𝑆‘𝑡) → 𝑡 ∈ 𝐽)
43	42	ad2antll 742	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝑡 ∈ 𝐽)
44		restopn2 23371	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝐽) → ((𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∈ (𝐽 ↾_t 𝑡) ↔ ((𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∈ 𝐽 ∧ (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ⊆ 𝑡)))
45	41, 43, 44	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → ((𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∈ (𝐽 ↾_t 𝑡) ↔ ((𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∈ 𝐽 ∧ (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ⊆ 𝑡)))
46	38, 45	mpbid 235	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → ((𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∈ 𝐽 ∧ (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ⊆ 𝑡))
47	46	simpld 500	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∈ 𝐽)
48	7	ffnd 6713	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → 𝐹 Fn 𝐵)
49	48	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝐹 Fn 𝐵)
50		inss1 4192	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ⊆ 𝐴
51	33, 10	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝐴 ⊆ 𝐵)
52	50, 51	sstrid 3951	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ⊆ 𝐵)
53		simplr 781	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝑧 ∈ 𝐴)
54	27	simprd 501	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝑧 ∈ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))
55	53, 54	elind 4156	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → 𝑧 ∈ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)))
56		fnfvima 7238	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐵 ∧ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ⊆ 𝐵 ∧ 𝑧 ∈ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) → (𝐹‘𝑧) ∈ (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))))
57	49, 52, 55, 56	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → (𝐹‘𝑧) ∈ (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))))
58		imass2 6109	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)) ⊆ 𝐴 → (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ⊆ (𝐹 “ 𝐴))
59	50, 58	mp1i 14	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ⊆ (𝐹 “ 𝐴))
60		eleq2 2855	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) → ((𝐹‘𝑧) ∈ 𝑦 ↔ (𝐹‘𝑧) ∈ (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥)))))
61		sseq1 3965	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) → (𝑦 ⊆ (𝐹 “ 𝐴) ↔ (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ⊆ (𝐹 “ 𝐴)))
62	60, 61	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) → (((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴)) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∧ (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ⊆ (𝐹 “ 𝐴))))
63	62	rspcev 3584	. . . . . . . . . . 11 ⊢ (((𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∈ 𝐽 ∧ ((𝐹‘𝑧) ∈ (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ∧ (𝐹 “ (𝐴 ∩ (℩𝑥 ∈ 𝑤 𝑧 ∈ 𝑥))) ⊆ (𝐹 “ 𝐴))) → ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴)))
64	47, 57, 59, 63	syl12anc 850	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑧) ∈ 𝑡 ∧ 𝑤 ∈ (𝑆‘𝑡))) → ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴)))
65	64	expr 462	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑧) ∈ 𝑡) → (𝑤 ∈ (𝑆‘𝑡) → ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴))))
66	65	exlimdv 1966	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑧) ∈ 𝑡) → (∃𝑤 𝑤 ∈ (𝑆‘𝑡) → ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴))))
67	17, 66	biimtrid 245	. . . . . . 7 ⊢ ((((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑧) ∈ 𝑡) → ((𝑆‘𝑡) ≠ ∅ → ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴))))
68	67	expimpd 459	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (((𝐹‘𝑧) ∈ 𝑡 ∧ (𝑆‘𝑡) ≠ ∅) → ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴))))
69	68	rexlimdvw 3174	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (∃𝑡 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑡 ∧ (𝑆‘𝑡) ≠ ∅) → ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴))))
70	16, 69	mpd 16	. . . 4 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴)))
71	70	ralrimiva 3160	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → ∀𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴)))
72		eleq1 2854	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥 ∈ 𝑦 ↔ (𝐹‘𝑧) ∈ 𝑦))
73	72	anbi1d 643	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴)) ↔ ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴))))
74	73	rexbidv 3192	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴)) ↔ ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴))))
75	74	ralima 7242	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ 𝐵) → (∀𝑥 ∈ (𝐹 “ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴)) ↔ ∀𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴))))
76	48, 11, 75	syl2anc 596	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → (∀𝑥 ∈ (𝐹 “ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴)) ↔ ∀𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐽 ((𝐹‘𝑧) ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴))))
77	71, 76	mpbird 260	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → ∀𝑥 ∈ (𝐹 “ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴)))
78		eltop2 23169	. . 3 ⊢ (𝐽 ∈ Top → ((𝐹 “ 𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝐹 “ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴))))
79	40, 78	syl 18	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → ((𝐹 “ 𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝐹 “ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝐹 “ 𝐴))))
80	77, 79	mpbird 260	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → (𝐹 “ 𝐴) ∈ 𝐽)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 ∃wrex 3092 {crab 3419 ∖ cdif 3905 ∩ cin 3907 ⊆ wss 3908 ∅c0 4289 𝒫 cpw 4567 {csn 4594 ∪ cuni 4877 ↦ cmpt 5197 ^◡ccnv 5665 ↾ cres 5668 “ cima 5669 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 ℩crio 7379 (class class class)co 7423 ↾_t crest 17498 Topctop 23087 Cn ccn 23418 Homeochmeo 23947 CovMap ccvm 35768

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-map 8835 df-en 8953 df-fin 8956 df-fi 9381 df-rest 17500 df-topgen 17521 df-top 23088 df-topon 23105 df-bases 23140 df-cn 23421 df-hmeo 23949 df-cvm 35769

This theorem is used by: cvmopn 35793

W3C validator