Theorem cnptopresti 14406

Description: One direction of cnptoprest 14407 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 31-Mar-2023.)

Assertion

Ref	Expression
cnptopresti	⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃))

Proof of Theorem cnptopresti

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 527	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐽 ∈ (TopOn‘𝑋))
2		toptopon2 14187	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
3	2	biimpi 120	. . . . 5 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘∪ 𝐾))
4	3	ad2antlr 489	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐾 ∈ (TopOn‘∪ 𝐾))
5		simpr3 1007	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
6		cnpf2 14375	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶∪ 𝐾)
7	1, 4, 5, 6	syl3anc 1249	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐹:𝑋⟶∪ 𝐾)
8		simpr1 1005	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐴 ⊆ 𝑋)
9	7, 8	fssresd 5430	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝐹 ↾ 𝐴):𝐴⟶∪ 𝐾)
10		simplr2 1042	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) → 𝑃 ∈ 𝐴)
11		fvres 5578	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑃) = (𝐹‘𝑃))
12	10, 11	syl 14	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) → ((𝐹 ↾ 𝐴)‘𝑃) = (𝐹‘𝑃))
13	12	eleq1d 2262	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) → (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 ↔ (𝐹‘𝑃) ∈ 𝑦))
14	1	ad2antrr 488	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) ∧ (𝐹‘𝑃) ∈ 𝑦) → 𝐽 ∈ (TopOn‘𝑋))
15	4	ad2antrr 488	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) ∧ (𝐹‘𝑃) ∈ 𝑦) → 𝐾 ∈ (TopOn‘∪ 𝐾))
16	8	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) ∧ (𝐹‘𝑃) ∈ 𝑦) → 𝐴 ⊆ 𝑋)
17		simpr2 1006	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝑃 ∈ 𝐴)
18	17	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) ∧ (𝐹‘𝑃) ∈ 𝑦) → 𝑃 ∈ 𝐴)
19	16, 18	sseldd 3180	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) ∧ (𝐹‘𝑃) ∈ 𝑦) → 𝑃 ∈ 𝑋)
20	5	ad2antrr 488	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) ∧ (𝐹‘𝑃) ∈ 𝑦) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
21		simplr 528	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) ∧ (𝐹‘𝑃) ∈ 𝑦) → 𝑦 ∈ 𝐾)
22		simpr 110	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) ∧ (𝐹‘𝑃) ∈ 𝑦) → (𝐹‘𝑃) ∈ 𝑦)
23		icnpimaex 14379	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))
24	14, 15, 19, 20, 21, 22, 23	syl33anc 1264	. . . . . 6 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) ∧ (𝐹‘𝑃) ∈ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))
25	24	ex 115	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))
26		idd 21	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝑃 ∈ 𝑥 → 𝑃 ∈ 𝑥))
27	26, 17	jctird 317	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝑃 ∈ 𝑥 → (𝑃 ∈ 𝑥 ∧ 𝑃 ∈ 𝐴)))
28		elin 3342	. . . . . . . . . 10 ⊢ (𝑃 ∈ (𝑥 ∩ 𝐴) ↔ (𝑃 ∈ 𝑥 ∧ 𝑃 ∈ 𝐴))
29	27, 28	imbitrrdi 162	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝑃 ∈ 𝑥 → 𝑃 ∈ (𝑥 ∩ 𝐴)))
30		inss1 3379	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
31		imass2 5041	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ (𝐹 “ 𝑥))
32	30, 31	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ (𝐹 “ 𝑥)
33		id 19	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐹 “ 𝑥) ⊆ 𝑦)
34	32, 33	sstrid 3190	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)
35	34	a1i 9	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦))
36	29, 35	anim12d 335	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → (𝑃 ∈ (𝑥 ∩ 𝐴) ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)))
37	36	reximdv 2595	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ (𝑥 ∩ 𝐴) ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)))
38		vex 2763	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
39	38	inex1 4163	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ∈ V
40	39	a1i 9	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ V)
41		topontop 14182	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
42	41	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐽 ∈ Top)
43		uniexg 4470	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
44	42, 43	syl 14	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → ∪ 𝐽 ∈ V)
45		toponuni 14183	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
46	45	sseq2d 3209	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝐽))
47	46	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝐽))
48	8, 47	mpbid 147	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐴 ⊆ ∪ 𝐽)
49	44, 48	ssexd 4169	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐴 ∈ V)
50		elrest 12857	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐴)))
51	42, 49, 50	syl2anc 411	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝑧 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐴)))
52		simpr 110	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → 𝑧 = (𝑥 ∩ 𝐴))
53	52	eleq2d 2263	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ (𝑥 ∩ 𝐴)))
54	52	imaeq2d 5005	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → ((𝐹 ↾ 𝐴) “ 𝑧) = ((𝐹 ↾ 𝐴) “ (𝑥 ∩ 𝐴)))
55		inss2 3380	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
56		resima2 4976	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝐴) ⊆ 𝐴 → ((𝐹 ↾ 𝐴) “ (𝑥 ∩ 𝐴)) = (𝐹 “ (𝑥 ∩ 𝐴)))
57	55, 56	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ 𝐴) “ (𝑥 ∩ 𝐴)) = (𝐹 “ (𝑥 ∩ 𝐴))
58	54, 57	eqtrdi 2242	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → ((𝐹 ↾ 𝐴) “ 𝑧) = (𝐹 “ (𝑥 ∩ 𝐴)))
59	58	sseq1d 3208	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → (((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦))
60	53, 59	anbi12d 473	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → ((𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥 ∩ 𝐴) ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)))
61	40, 51, 60	rexxfr2d 4496	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ (𝑥 ∩ 𝐴) ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)))
62	37, 61	sylibrd 169	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))
63	62	adantr 276	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))
64	25, 63	syld 45	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))
65	13, 64	sylbid 150	. . 3 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦 ∈ 𝐾) → (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))
66	65	ralrimiva 2567	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))
67		resttopon 14339	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
68	1, 8, 67	syl2anc 411	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
69		iscnp 14367	. . 3 ⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ 𝐴) → ((𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹 ↾ 𝐴):𝐴⟶∪ 𝐾 ∧ ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))))
70	68, 4, 17, 69	syl3anc 1249	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → ((𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹 ↾ 𝐴):𝐴⟶∪ 𝐾 ∧ ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))))
71	9, 66, 70	mpbir2and 946	1 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  Vcvv 2760   ∩ cin 3152   ⊆ wss 3153  ∪ cuni 3835   ↾ cres 4661   “ cima 4662  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918   ↾_t crest 12850  Topctop 14165  TopOnctopon 14178   CnP ccnp 14354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-rest 12852  df-topgen 12871  df-top 14166  df-topon 14179  df-bases 14211  df-cnp 14357

This theorem is referenced by: (None)