Theorem 1stcrest 23316

Description: A subspace of a first-countable space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
1stcrest	⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ 1stω)

Proof of Theorem 1stcrest

Dummy variables 𝑡 𝑎 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1stctop 23306	. . 3 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)
2		resttop 23023	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Top)
3	1, 2	sylan 580	. 2 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Top)
4		eqid 2729	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	restuni2 23030	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ ∪ 𝐽) = ∪ (𝐽 ↾t 𝐴))
6	1, 5	sylan 580	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ ∪ 𝐽) = ∪ (𝐽 ↾t 𝐴))
7	6	eleq2d 2814	. . . . 5 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (𝐴 ∩ ∪ 𝐽) ↔ 𝑥 ∈ ∪ (𝐽 ↾t 𝐴)))
8	7	biimpar 477	. . . 4 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ ∪ (𝐽 ↾t 𝐴)) → 𝑥 ∈ (𝐴 ∩ ∪ 𝐽))
9		simpl 482	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ 1stω)
10		elinel2 4161	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽)
11	4	1stcclb 23307	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))
12	9, 10, 11	syl2an 596	. . . . 5 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) → ∃𝑡 ∈ 𝒫 𝐽(𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))
13		simplll 774	. . . . . . . 8 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → 𝐽 ∈ 1stω)
14		elpwi 4566	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝒫 𝐽 → 𝑡 ⊆ 𝐽)
15	14	ad2antrl 728	. . . . . . . 8 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → 𝑡 ⊆ 𝐽)
16		ssrest 23039	. . . . . . . 8 ⊢ ((𝐽 ∈ 1stω ∧ 𝑡 ⊆ 𝐽) → (𝑡 ↾t 𝐴) ⊆ (𝐽 ↾t 𝐴))
17	13, 15, 16	syl2anc 584	. . . . . . 7 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → (𝑡 ↾t 𝐴) ⊆ (𝐽 ↾t 𝐴))
18		ovex 7402	. . . . . . . 8 ⊢ (𝐽 ↾t 𝐴) ∈ V
19	18	elpw2 5284	. . . . . . 7 ⊢ ((𝑡 ↾t 𝐴) ∈ 𝒫 (𝐽 ↾t 𝐴) ↔ (𝑡 ↾t 𝐴) ⊆ (𝐽 ↾t 𝐴))
20	17, 19	sylibr 234	. . . . . 6 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → (𝑡 ↾t 𝐴) ∈ 𝒫 (𝐽 ↾t 𝐴))
21		vex 3448	. . . . . . . 8 ⊢ 𝑡 ∈ V
22		simpllr 775	. . . . . . . 8 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → 𝐴 ∈ 𝑉)
23		restval 17365	. . . . . . . 8 ⊢ ((𝑡 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑡 ↾t 𝐴) = ran (𝑣 ∈ 𝑡 ↦ (𝑣 ∩ 𝐴)))
24	21, 22, 23	sylancr 587	. . . . . . 7 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → (𝑡 ↾t 𝐴) = ran (𝑣 ∈ 𝑡 ↦ (𝑣 ∩ 𝐴)))
25		simprrl 780	. . . . . . . 8 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → 𝑡 ≼ ω)
26		1stcrestlem 23315	. . . . . . . 8 ⊢ (𝑡 ≼ ω → ran (𝑣 ∈ 𝑡 ↦ (𝑣 ∩ 𝐴)) ≼ ω)
27	25, 26	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → ran (𝑣 ∈ 𝑡 ↦ (𝑣 ∩ 𝐴)) ≼ ω)
28	24, 27	eqbrtrd 5124	. . . . . 6 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → (𝑡 ↾t 𝐴) ≼ ω)
29	1	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → 𝐽 ∈ Top)
30		elrest 17366	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑧 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑎 ∈ 𝐽 𝑧 = (𝑎 ∩ 𝐴)))
31	29, 22, 30	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → (𝑧 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑎 ∈ 𝐽 𝑧 = (𝑎 ∩ 𝐴)))
32		r19.29 3094	. . . . . . . . . . . 12 ⊢ ((∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎)) ∧ ∃𝑎 ∈ 𝐽 𝑧 = (𝑎 ∩ 𝐴)) → ∃𝑎 ∈ 𝐽 ((𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎)) ∧ 𝑧 = (𝑎 ∩ 𝐴)))
33		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
34	33	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝐴))
35	34	ancld 550	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑦 ∧ 𝑥 ∈ 𝐴)))
36		elin 3927	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (𝑦 ∩ 𝐴) ↔ (𝑥 ∈ 𝑦 ∧ 𝑥 ∈ 𝐴))
37	35, 36	imbitrrdi 252	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝑦 → 𝑥 ∈ (𝑦 ∩ 𝐴)))
38		ssrin 4201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ⊆ 𝑎 → (𝑦 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴))
39	37, 38	anim12d1 610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎) → (𝑥 ∈ (𝑦 ∩ 𝐴) ∧ (𝑦 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴))))
40	39	reximdv 3148	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → (∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎) → ∃𝑦 ∈ 𝑡 (𝑥 ∈ (𝑦 ∩ 𝐴) ∧ (𝑦 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴))))
41		vex 3448	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑦 ∈ V
42	41	inex1 5267	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∩ 𝐴) ∈ V
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ 𝑡) → (𝑦 ∩ 𝐴) ∈ V)
44		simp-4r 783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → 𝐴 ∈ 𝑉)
45		elrest 17366	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑤 ∈ (𝑡 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑡 𝑤 = (𝑦 ∩ 𝐴)))
46	21, 44, 45	sylancr 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → (𝑤 ∈ (𝑡 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑡 𝑤 = (𝑦 ∩ 𝐴)))
47		eleq2 2817	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (𝑦 ∩ 𝐴) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ (𝑦 ∩ 𝐴)))
48		sseq1 3969	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (𝑦 ∩ 𝐴) → (𝑤 ⊆ (𝑎 ∩ 𝐴) ↔ (𝑦 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴)))
49	47, 48	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (𝑦 ∩ 𝐴) → ((𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴)) ↔ (𝑥 ∈ (𝑦 ∩ 𝐴) ∧ (𝑦 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴))))
50	49	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) ∧ 𝑤 = (𝑦 ∩ 𝐴)) → ((𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴)) ↔ (𝑥 ∈ (𝑦 ∩ 𝐴) ∧ (𝑦 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴))))
51	43, 46, 50	rexxfr2d 5361	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → (∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴)) ↔ ∃𝑦 ∈ 𝑡 (𝑥 ∈ (𝑦 ∩ 𝐴) ∧ (𝑦 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴))))
52	40, 51	sylibrd 259	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → (∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎) → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴))))
53	52	expr 456	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎 ∈ 𝐽) → (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎) → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴)))))
54	53	com23 86	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎 ∈ 𝐽) → (∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎) → (𝑥 ∈ 𝐴 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴)))))
55	54	imim2d 57	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎 ∈ 𝐽) → ((𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎)) → (𝑥 ∈ 𝑎 → (𝑥 ∈ 𝐴 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴))))))
56	55	imp4b 421	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎 ∈ 𝐽) ∧ (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))) → ((𝑥 ∈ 𝑎 ∧ 𝑥 ∈ 𝐴) → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴))))
57		eleq2 2817	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑎 ∩ 𝐴) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑎 ∩ 𝐴)))
58		elin 3927	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐴) ↔ (𝑥 ∈ 𝑎 ∧ 𝑥 ∈ 𝐴))
59	57, 58	bitrdi 287	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑎 ∩ 𝐴) → (𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑎 ∧ 𝑥 ∈ 𝐴)))
60		sseq2 3970	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑎 ∩ 𝐴) → (𝑤 ⊆ 𝑧 ↔ 𝑤 ⊆ (𝑎 ∩ 𝐴)))
61	60	anbi2d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑎 ∩ 𝐴) → ((𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴))))
62	61	rexbidv 3157	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑎 ∩ 𝐴) → (∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴))))
63	59, 62	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑎 ∩ 𝐴) → ((𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ ((𝑥 ∈ 𝑎 ∧ 𝑥 ∈ 𝐴) → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴)))))
64	56, 63	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ ((((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎 ∈ 𝐽) ∧ (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))) → (𝑧 = (𝑎 ∩ 𝐴) → (𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
65	64	expimpd 453	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ 𝑎 ∈ 𝐽) → (((𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎)) ∧ 𝑧 = (𝑎 ∩ 𝐴)) → (𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
66	65	rexlimdva 3134	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → (∃𝑎 ∈ 𝐽 ((𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎)) ∧ 𝑧 = (𝑎 ∩ 𝐴)) → (𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
67	32, 66	syl5 34	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → ((∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎)) ∧ ∃𝑎 ∈ 𝐽 𝑧 = (𝑎 ∩ 𝐴)) → (𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
68	67	expd 415	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) → (∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎)) → (∃𝑎 ∈ 𝐽 𝑧 = (𝑎 ∩ 𝐴) → (𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))))
69	68	impr 454	. . . . . . . . 9 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎)))) → (∃𝑎 ∈ 𝐽 𝑧 = (𝑎 ∩ 𝐴) → (𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
70	69	adantrrl 724	. . . . . . . 8 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → (∃𝑎 ∈ 𝐽 𝑧 = (𝑎 ∩ 𝐴) → (𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
71	31, 70	sylbid 240	. . . . . . 7 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → (𝑧 ∈ (𝐽 ↾t 𝐴) → (𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
72	71	ralrimiv 3124	. . . . . 6 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → ∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
73		breq1 5105	. . . . . . . 8 ⊢ (𝑦 = (𝑡 ↾t 𝐴) → (𝑦 ≼ ω ↔ (𝑡 ↾t 𝐴) ≼ ω))
74		rexeq 3292	. . . . . . . . . 10 ⊢ (𝑦 = (𝑡 ↾t 𝐴) → (∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
75	74	imbi2d 340	. . . . . . . . 9 ⊢ (𝑦 = (𝑡 ↾t 𝐴) → ((𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ (𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
76	75	ralbidv 3156	. . . . . . . 8 ⊢ (𝑦 = (𝑡 ↾t 𝐴) → (∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ ∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
77	73, 76	anbi12d 632	. . . . . . 7 ⊢ (𝑦 = (𝑡 ↾t 𝐴) → ((𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ↔ ((𝑡 ↾t 𝐴) ≼ ω ∧ ∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))))
78	77	rspcev 3585	. . . . . 6 ⊢ (((𝑡 ↾t 𝐴) ∈ 𝒫 (𝐽 ↾t 𝐴) ∧ ((𝑡 ↾t 𝐴) ≼ ω ∧ ∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) → ∃𝑦 ∈ 𝒫 (𝐽 ↾t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
79	20, 28, 72, 78	syl12anc 836	. . . . 5 ⊢ ((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ (𝑡 ∈ 𝒫 𝐽 ∧ (𝑡 ≼ ω ∧ ∀𝑎 ∈ 𝐽 (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑡 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑎))))) → ∃𝑦 ∈ 𝒫 (𝐽 ↾t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
80	12, 79	rexlimddv 3140	. . . 4 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) → ∃𝑦 ∈ 𝒫 (𝐽 ↾t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
81	8, 80	syldan 591	. . 3 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ ∪ (𝐽 ↾t 𝐴)) → ∃𝑦 ∈ 𝒫 (𝐽 ↾t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
82	81	ralrimiva 3125	. 2 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ ∪ (𝐽 ↾t 𝐴)∃𝑦 ∈ 𝒫 (𝐽 ↾t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
83		eqid 2729	. . 3 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴)
84	83	is1stc2 23305	. 2 ⊢ ((𝐽 ↾t 𝐴) ∈ 1stω ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ ∀𝑥 ∈ ∪ (𝐽 ↾t 𝐴)∃𝑦 ∈ 𝒫 (𝐽 ↾t 𝐴)(𝑦 ≼ ω ∧ ∀𝑧 ∈ (𝐽 ↾t 𝐴)(𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))))
85	3, 82, 84	sylanbrc 583	1 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ 1stω)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867   class class class wbr 5102   ↦ cmpt 5183  ran crn 5632  (class class class)co 7369  ωcom 7822   ≼ cdom 8893   ↾_t crest 17359  Topctop 22756  1^stωc1stc 23300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-fin 8899  df-fi 9338  df-card 9868  df-acn 9871  df-rest 17361  df-topgen 17382  df-top 22757  df-topon 22774  df-bases 22809  df-1stc 23302

This theorem is referenced by:  lly1stc  23359