Theorem 1stcrest 23771

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 23761	. . 3 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)
2		resttop 23478	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Top)
3	1, 2	sylan 592	. 2 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Top)
4		eqid 2761	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	restuni2 23485	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ ∪ 𝐽) = ∪ (𝐽 ↾t 𝐴))
6	1, 5	sylan 592	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ ∪ 𝐽) = ∪ (𝐽 ↾t 𝐴))
7	6	eleq2d 2847	. . . . 5 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (𝐴 ∩ ∪ 𝐽) ↔ 𝑥 ∈ ∪ (𝐽 ↾t 𝐴)))
8	7	biimpar 483	. . . 4 ⊢ (((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ ∪ (𝐽 ↾t 𝐴)) → 𝑥 ∈ (𝐴 ∩ ∪ 𝐽))
9		simpl 488	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ 1stω)
10		elinel2 4148	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽)
11	4	1stcclb 23762	. . . . . 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 608	. . . . 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 787	. . . . . . . 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 4564	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝒫 𝐽 → 𝑡 ⊆ 𝐽)
15	14	ad2antrl 741	. . . . . . . 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 23494	. . . . . . . 8 ⊢ ((𝐽 ∈ 1stω ∧ 𝑡 ⊆ 𝐽) → (𝑡 ↾t 𝐴) ⊆ (𝐽 ↾t 𝐴))
17	13, 15, 16	syl2anc 596	. . . . . . 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 7453	. . . . . . . 8 ⊢ (𝐽 ↾t 𝐴) ∈ V
19	18	elpw2 5296	. . . . . . 7 ⊢ ((𝑡 ↾t 𝐴) ∈ 𝒫 (𝐽 ↾t 𝐴) ↔ (𝑡 ↾t 𝐴) ⊆ (𝐽 ↾t 𝐴))
20	17, 19	sylibr 237	. . . . . 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 3455	. . . . . . . 8 ⊢ 𝑡 ∈ V
22		simpllr 788	. . . . . . . 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 17597	. . . . . . . 8 ⊢ ((𝑡 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑡 ↾t 𝐴) = ran (𝑣 ∈ 𝑡 ↦ (𝑣 ∩ 𝐴)))
24	21, 22, 23	sylancr 599	. . . . . . 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 793	. . . . . . . 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 23770	. . . . . . . 8 ⊢ (𝑡 ≼ ω → ran (𝑣 ∈ 𝑡 ↦ (𝑣 ∩ 𝐴)) ≼ ω)
27	25, 26	syl 18	. . . . . . 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 5127	. . . . . 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 743	. . . . . . . . 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 17598	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑧 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑎 ∈ 𝐽 𝑧 = (𝑎 ∩ 𝐴)))
31	29, 22, 30	syl2anc 596	. . . . . . . 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 3126	. . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
34	33	a1d 26	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝐴))
35	34	ancld 560	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑦 ∧ 𝑥 ∈ 𝐴)))
36		elin 3915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (𝑦 ∩ 𝐴) ↔ (𝑥 ∈ 𝑦 ∧ 𝑥 ∈ 𝐴))
37	35, 36	imbitrrdi 255	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝑦 → 𝑥 ∈ (𝑦 ∩ 𝐴)))
38		ssrin 4187	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ⊆ 𝑎 → (𝑦 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴))
39	37, 38	anim12d1 622	. . . . . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . . . . . 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 3455	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑦 ∈ V
42	41	inex1 5277	. . . . . . . . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ (𝐴 ∩ ∪ 𝐽)) ∧ 𝑡 ∈ 𝒫 𝐽) ∧ (𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝐴)) → 𝐴 ∈ 𝑉)
45		elrest 17598	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑤 ∈ (𝑡 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑡 𝑤 = (𝑦 ∩ 𝐴)))
46	21, 44, 45	sylancr 599	. . . . . . . . . . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (𝑦 ∩ 𝐴) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ (𝑦 ∩ 𝐴)))
48		sseq1 3956	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (𝑦 ∩ 𝐴) → (𝑤 ⊆ (𝑎 ∩ 𝐴) ↔ (𝑦 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴)))
49	47, 48	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (𝑦 ∩ 𝐴) → ((𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴)) ↔ (𝑥 ∈ (𝑦 ∩ 𝐴) ∧ (𝑦 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴))))
50	49	adantl 487	. . . . . . . . . . . . . . . . . . . . 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 5373	. . . . . . . . . . . . . . . . . . . 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 262	. . . . . . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . 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 58	. . . . . . . . . . . . . . . 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 427	. . . . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑎 ∩ 𝐴) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑎 ∩ 𝐴)))
58		elin 3915	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑎 ∩ 𝐴) ↔ (𝑥 ∈ 𝑎 ∧ 𝑥 ∈ 𝐴))
59	57, 58	bitrdi 290	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑎 ∩ 𝐴) → (𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑎 ∧ 𝑥 ∈ 𝐴)))
60		sseq2 3957	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑎 ∩ 𝐴) → (𝑤 ⊆ 𝑧 ↔ 𝑤 ⊆ (𝑎 ∩ 𝐴)))
61	60	anbi2d 642	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑎 ∩ 𝐴) → ((𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑎 ∩ 𝐴))))
62	61	rexbidv 3187	. . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . 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 250	. . . . . . . . . . . . . 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 459	. . . . . . . . . . . . 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 3164	. . . . . . . . . . . 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 35	. . . . . . . . . . 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 421	. . . . . . . . . 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 460	. . . . . . . . 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 737	. . . . . . . 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 243	. . . . . . 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 3154	. . . . . 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 5106	. . . . . . . 8 ⊢ (𝑦 = (𝑡 ↾t 𝐴) → (𝑦 ≼ ω ↔ (𝑡 ↾t 𝐴) ≼ ω))
74		rexeq 3316	. . . . . . . . . 10 ⊢ (𝑦 = (𝑡 ↾t 𝐴) → (∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤 ∈ (𝑡 ↾t 𝐴)(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
75	74	imbi2d 343	. . . . . . . . 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 3186	. . . . . . . 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 644	. . . . . . 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 3577	. . . . . 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 850	. . . . 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 3170	. . . 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 603	. . 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 3155	. 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 2761	. . 3 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴)
84	83	is1stc2 23760	. 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 595	1 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ 1stω)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867   class class class wbr 5103   ↦ cmpt 5186  ran crn 5652  (class class class)co 7420  ωcom 7877   ≼ cdom 8971   ↾_t crest 17591  Topctop 23211  1^stωc1stc 23755

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-fin 8977  df-fi 9403  df-card 10020  df-acn 10023  df-rest 17593  df-topgen 17614  df-top 23212  df-topon 23229  df-bases 23264  df-1stc 23757

This theorem is used by:  lly1stc  23815