Theorem cnptoprest2 12781

Description: Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)

Hypotheses

Ref	Expression
cnprest.1	⊢ 𝑋 = ∪ 𝐽
cnprest.2	⊢ 𝑌 = ∪ 𝐾

Assertion

Ref	Expression
cnptoprest2	⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)))

Proof of Theorem cnptoprest2

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

Step	Hyp	Ref	Expression
1		cnprest.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
2	1	toptopon 12557	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3	2	biimpi 119	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
4	3	ad2antrr 480	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
5	4	adantr 274	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
6		simplr 520	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐾 ∈ Top)
7	6	adantr 274	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
8		simpr 109	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
9		cnprcl2k 12747	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋)
10	5, 7, 8, 9	syl3anc 1227	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋)
11	10	ex 114	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ 𝑋))
12	4	adantr 274	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
13		cnprest.2	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐾
14		uniexg 4411	. . . . . . . . 9 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
15	13, 14	eqeltrid 2251	. . . . . . . 8 ⊢ (𝐾 ∈ Top → 𝑌 ∈ V)
16	6, 15	syl 14	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝑌 ∈ V)
17		simprr 522	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ⊆ 𝑌)
18	16, 17	ssexd 4116	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ∈ V)
19		resttop 12711	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ V) → (𝐾 ↾t 𝐵) ∈ Top)
20	6, 18, 19	syl2anc 409	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐾 ↾t 𝐵) ∈ Top)
21	20	adantr 274	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → (𝐾 ↾t 𝐵) ∈ Top)
22		simpr 109	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃))
23		cnprcl2k 12747	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐾 ↾t 𝐵) ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → 𝑃 ∈ 𝑋)
24	12, 21, 22, 23	syl3anc 1227	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → 𝑃 ∈ 𝑋)
25	24	ex 114	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) → 𝑃 ∈ 𝑋))
26		simprl 521	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐹:𝑋⟶𝐵)
27	26	ffvelrnda 5614	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹‘𝑃) ∈ 𝐵)
28	27	biantrud 302	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹‘𝑃) ∈ 𝑥 ↔ ((𝐹‘𝑃) ∈ 𝑥 ∧ (𝐹‘𝑃) ∈ 𝐵)))
29		elin 3300	. . . . . . . 8 ⊢ ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) ↔ ((𝐹‘𝑃) ∈ 𝑥 ∧ (𝐹‘𝑃) ∈ 𝐵))
30	28, 29	bitr4di 197	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹‘𝑃) ∈ 𝑥 ↔ (𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵)))
31		imassrn 4951	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝑦) ⊆ ran 𝐹
32		simplrl 525	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐹:𝑋⟶𝐵)
33	32	frnd 5341	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ran 𝐹 ⊆ 𝐵)
34	31, 33	sstrid 3148	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 “ 𝑦) ⊆ 𝐵)
35	34	biantrud 302	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ ((𝐹 “ 𝑦) ⊆ 𝑥 ∧ (𝐹 “ 𝑦) ⊆ 𝐵)))
36		ssin 3339	. . . . . . . . . 10 ⊢ (((𝐹 “ 𝑦) ⊆ 𝑥 ∧ (𝐹 “ 𝑦) ⊆ 𝐵) ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))
37	35, 36	bitrdi 195	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))
38	37	anbi2d 460	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥) ↔ (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))
39	38	rexbidv 2465	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥) ↔ ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))
40	30, 39	imbi12d 233	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)) ↔ ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))))
41	40	ralbidv 2464	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)) ↔ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))))
42		vex 2724	. . . . . . . 8 ⊢ 𝑥 ∈ V
43	42	inex1 4110	. . . . . . 7 ⊢ (𝑥 ∩ 𝐵) ∈ V
44	43	a1i 9	. . . . . 6 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) ∧ 𝑥 ∈ 𝐾) → (𝑥 ∩ 𝐵) ∈ V)
45	6	adantr 274	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐾 ∈ Top)
46	18	adantr 274	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐵 ∈ V)
47		elrest 12499	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐾 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑧 = (𝑥 ∩ 𝐵)))
48	45, 46, 47	syl2anc 409	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝑧 ∈ (𝐾 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑧 = (𝑥 ∩ 𝐵)))
49		eleq2 2228	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝐹‘𝑃) ∈ 𝑧 ↔ (𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵)))
50		sseq2 3161	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝐹 “ 𝑦) ⊆ 𝑧 ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))
51	50	anbi2d 460	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))
52	51	rexbidv 2465	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∩ 𝐵) → (∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧) ↔ ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))
53	49, 52	imbi12d 233	. . . . . . 7 ⊢ (𝑧 = (𝑥 ∩ 𝐵) → (((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)) ↔ ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))))
54	53	adantl 275	. . . . . 6 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → (((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)) ↔ ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))))
55	44, 48, 54	ralxfr2d 4436	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)) ↔ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))))
56	41, 55	bitr4d 190	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)) ↔ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧))))
57	4	adantr 274	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
58	13	toptopon 12557	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
59	45, 58	sylib 121	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
60		simpr 109	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
61		iscnp 12740	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)))))
62	57, 59, 60, 61	syl3anc 1227	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)))))
63	17	adantr 274	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐵 ⊆ 𝑌)
64	32, 63	fssd 5344	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐹:𝑋⟶𝑌)
65	64	biantrurd 303	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)))))
66	62, 65	bitr4d 190	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))))
67		resttopon 12712	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ⊆ 𝑌) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
68	59, 63, 67	syl2anc 409	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
69		iscnp 12740	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) ↔ (𝐹:𝑋⟶𝐵 ∧ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)))))
70	57, 68, 60, 69	syl3anc 1227	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) ↔ (𝐹:𝑋⟶𝐵 ∧ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)))))
71	26	biantrurd 303	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)) ↔ (𝐹:𝑋⟶𝐵 ∧ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)))))
72	71	adantr 274	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)) ↔ (𝐹:𝑋⟶𝐵 ∧ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)))))
73	70, 72	bitr4d 190	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) ↔ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧))))
74	56, 66, 73	3bitr4d 219	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)))
75	74	ex 114	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝑃 ∈ 𝑋 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃))))
76	11, 25, 75	pm5.21ndd 695	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1342   ∈ wcel 2135  ∀wral 2442  ∃wrex 2443  Vcvv 2721   ∩ cin 3110   ⊆ wss 3111  ∪ cuni 3783  ran crn 4599   “ cima 4601  ⟶wf 5178  ‘cfv 5182  (class class class)co 5836   ↾_t crest 12492  Topctop 12536  TopOnctopon 12549   CnP ccnp 12727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-map 6607  df-rest 12494  df-topgen 12513  df-top 12537  df-topon 12550  df-bases 12582  df-cnp 12730

This theorem is referenced by: (None)