Theorem cnptoprest2 14712

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 14490	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3	2	biimpi 120	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
4	3	ad2antrr 488	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
5	4	adantr 276	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
6		simplr 528	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐾 ∈ Top)
7	6	adantr 276	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
8		simpr 110	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
9		cnprcl2k 14678	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋)
10	5, 7, 8, 9	syl3anc 1250	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋)
11	10	ex 115	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ 𝑋))
12	4	adantr 276	. . . 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 4486	. . . . . . . . 9 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
15	13, 14	eqeltrid 2292	. . . . . . . 8 ⊢ (𝐾 ∈ Top → 𝑌 ∈ V)
16	6, 15	syl 14	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝑌 ∈ V)
17		simprr 531	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ⊆ 𝑌)
18	16, 17	ssexd 4184	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ∈ V)
19		resttop 14642	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ V) → (𝐾 ↾t 𝐵) ∈ Top)
20	6, 18, 19	syl2anc 411	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐾 ↾t 𝐵) ∈ Top)
21	20	adantr 276	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → (𝐾 ↾t 𝐵) ∈ Top)
22		simpr 110	. . . 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 14678	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐾 ↾t 𝐵) ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → 𝑃 ∈ 𝑋)
24	12, 21, 22, 23	syl3anc 1250	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → 𝑃 ∈ 𝑋)
25	24	ex 115	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) → 𝑃 ∈ 𝑋))
26		simprl 529	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐹:𝑋⟶𝐵)
27	26	ffvelcdmda 5715	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹‘𝑃) ∈ 𝐵)
28	27	biantrud 304	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹‘𝑃) ∈ 𝑥 ↔ ((𝐹‘𝑃) ∈ 𝑥 ∧ (𝐹‘𝑃) ∈ 𝐵)))
29		elin 3356	. . . . . . . 8 ⊢ ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) ↔ ((𝐹‘𝑃) ∈ 𝑥 ∧ (𝐹‘𝑃) ∈ 𝐵))
30	28, 29	bitr4di 198	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹‘𝑃) ∈ 𝑥 ↔ (𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵)))
31		imassrn 5033	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝑦) ⊆ ran 𝐹
32		simplrl 535	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐹:𝑋⟶𝐵)
33	32	frnd 5435	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ran 𝐹 ⊆ 𝐵)
34	31, 33	sstrid 3204	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 “ 𝑦) ⊆ 𝐵)
35	34	biantrud 304	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ ((𝐹 “ 𝑦) ⊆ 𝑥 ∧ (𝐹 “ 𝑦) ⊆ 𝐵)))
36		ssin 3395	. . . . . . . . . 10 ⊢ (((𝐹 “ 𝑦) ⊆ 𝑥 ∧ (𝐹 “ 𝑦) ⊆ 𝐵) ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))
37	35, 36	bitrdi 196	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))
38	37	anbi2d 464	. . . . . . . 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 2507	. . . . . . 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 234	. . . . . 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 2506	. . . . 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 2775	. . . . . . . 8 ⊢ 𝑥 ∈ V
43	42	inex1 4178	. . . . . . 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 276	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐾 ∈ Top)
46	18	adantr 276	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐵 ∈ V)
47		elrest 13078	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐾 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑧 = (𝑥 ∩ 𝐵)))
48	45, 46, 47	syl2anc 411	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝑧 ∈ (𝐾 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑧 = (𝑥 ∩ 𝐵)))
49		eleq2 2269	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝐹‘𝑃) ∈ 𝑧 ↔ (𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵)))
50		sseq2 3217	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝐹 “ 𝑦) ⊆ 𝑧 ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))
51	50	anbi2d 464	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))
52	51	rexbidv 2507	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∩ 𝐵) → (∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧) ↔ ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))
53	49, 52	imbi12d 234	. . . . . . 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 277	. . . . . 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 4511	. . . . 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 191	. . . 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 276	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
58	13	toptopon 14490	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
59	45, 58	sylib 122	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
60		simpr 110	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
61		iscnp 14671	. . . . . 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 1250	. . . . 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 276	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐵 ⊆ 𝑌)
64	32, 63	fssd 5438	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐹:𝑋⟶𝑌)
65	64	biantrurd 305	. . . . 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 191	. . . 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 14643	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ⊆ 𝑌) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
68	59, 63, 67	syl2anc 411	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
69		iscnp 14671	. . . . . 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 1250	. . . . 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 305	. . . . . 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 276	. . . . 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 191	. . . 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 220	. . 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 115	. 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 707	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 104   ↔ wb 105   = wceq 1373   ∈ wcel 2176  ∀wral 2484  ∃wrex 2485  Vcvv 2772   ∩ cin 3165   ⊆ wss 3166  ∪ cuni 3850  ran crn 4676   “ cima 4678  ⟶wf 5267  ‘cfv 5271  (class class class)co 5944   ↾_t crest 13071  Topctop 14469  TopOnctopon 14482   CnP ccnp 14658

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-rest 13073  df-topgen 13092  df-top 14470  df-topon 14483  df-bases 14515  df-cnp 14661

This theorem is referenced by: (None)