Theorem cnrest2 23659

Description: Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
cnrest2	⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))))

Proof of Theorem cnrest2

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

Step	Hyp	Ref	Expression
1		cntop1 23613	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2	1	a1i 11	. . 3 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top))
3		eqid 2763	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2763	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 23619	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	5	ffnd 6740	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 Fn ∪ 𝐽)
7	6	a1i 11	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 Fn ∪ 𝐽))
8		simp2 1155	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → ran 𝐹 ⊆ 𝐵)
9	7, 8	jctird 536	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 ⊆ 𝐵)))
10		df-f 6570	. . . 4 ⊢ (𝐹:∪ 𝐽⟶𝐵 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 ⊆ 𝐵))
11	9, 10	imbitrrdi 255	. . 3 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝐵))
12	2, 11	jcad 522	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)))
13		cntop1 23613	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵)) → 𝐽 ∈ Top)
14	13	adantl 487	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → 𝐽 ∈ Top)
15		toptopon2 23291	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
16	14, 15	sylib 221	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → 𝐽 ∈ (TopOn‘∪ 𝐽))
17		resttopon 23534	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ⊆ 𝑌) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
18	17	3adant2 1149	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
19	18	adantr 486	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
20		simpr 490	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵)))
21		cnf2 23622	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → 𝐹:∪ 𝐽⟶𝐵)
22	16, 19, 20, 21	syl3anc 1399	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → 𝐹:∪ 𝐽⟶𝐵)
23	14, 22	jca 521	. . 3 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵))
24	23	ex 418	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵)) → (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)))
25		vex 3458	. . . . . . . . 9 ⊢ 𝑥 ∈ V
26	25	inex1 5284	. . . . . . . 8 ⊢ (𝑥 ∩ 𝐵) ∈ V
27	26	a1i 11	. . . . . . 7 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → (𝑥 ∩ 𝐵) ∈ V)
28		simpl1 1210	. . . . . . . 8 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐾 ∈ (TopOn‘𝑌))
29		toponmax 23299	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
30	28, 29	syl 18	. . . . . . . . 9 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝑌 ∈ 𝐾)
31		simpl3 1212	. . . . . . . . 9 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐵 ⊆ 𝑌)
32	30, 31	ssexd 5293	. . . . . . . 8 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐵 ∈ V)
33		elrest 17648	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ V) → (𝑦 ∈ (𝐾 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑦 = (𝑥 ∩ 𝐵)))
34	28, 32, 33	syl2anc 596	. . . . . . 7 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → (𝑦 ∈ (𝐾 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑦 = (𝑥 ∩ 𝐵)))
35		imaeq2 6066	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (𝑥 ∩ 𝐵)))
36	35	eleq1d 2849	. . . . . . . 8 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((◡𝐹 “ 𝑦) ∈ 𝐽 ↔ (◡𝐹 “ (𝑥 ∩ 𝐵)) ∈ 𝐽))
37	36	adantl 487	. . . . . . 7 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑦 = (𝑥 ∩ 𝐵)) → ((◡𝐹 “ 𝑦) ∈ 𝐽 ↔ (◡𝐹 “ (𝑥 ∩ 𝐵)) ∈ 𝐽))
38	27, 34, 37	ralxfr2d 5379	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → (∀𝑦 ∈ (𝐾 ↾t 𝐵)(◡𝐹 “ 𝑦) ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐾 (◡𝐹 “ (𝑥 ∩ 𝐵)) ∈ 𝐽))
39		simplrr 790	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → 𝐹:∪ 𝐽⟶𝐵)
40		ffun 6742	. . . . . . . . . 10 ⊢ (𝐹:∪ 𝐽⟶𝐵 → Fun 𝐹)
41		inpreima 7095	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑥 ∩ 𝐵)) = ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝐵)))
42	39, 40, 41	3syl 19	. . . . . . . . 9 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ (𝑥 ∩ 𝐵)) = ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝐵)))
43		cnvimass 6217	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
44		cnvimarndm 6098	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
45	43, 44	sseqtrri 3962	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ ran 𝐹)
46		simpll2 1232	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → ran 𝐹 ⊆ 𝐵)
47		imass2 6073	. . . . . . . . . . . 12 ⊢ (ran 𝐹 ⊆ 𝐵 → (◡𝐹 “ ran 𝐹) ⊆ (◡𝐹 “ 𝐵))
48	46, 47	syl 18	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ ran 𝐹) ⊆ (◡𝐹 “ 𝐵))
49	45, 48	sstrid 3924	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝐵))
50		dfss2 4097	. . . . . . . . . 10 ⊢ ((◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝐵) ↔ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝐵)) = (◡𝐹 “ 𝑥))
51	49, 50	sylib 221	. . . . . . . . 9 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝐵)) = (◡𝐹 “ 𝑥))
52	42, 51	eqtrd 2798	. . . . . . . 8 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ (𝑥 ∩ 𝐵)) = (◡𝐹 “ 𝑥))
53	52	eleq1d 2849	. . . . . . 7 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → ((◡𝐹 “ (𝑥 ∩ 𝐵)) ∈ 𝐽 ↔ (◡𝐹 “ 𝑥) ∈ 𝐽))
54	53	ralbidva 3187	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → (∀𝑥 ∈ 𝐾 (◡𝐹 “ (𝑥 ∩ 𝐵)) ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))
55		simprr 785	. . . . . . . 8 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐹:∪ 𝐽⟶𝐵)
56	55, 31	fssd 6757	. . . . . . 7 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐹:∪ 𝐽⟶𝑌)
57	56	biantrurd 542	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → (∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (𝐹:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
58	38, 54, 57	3bitrrd 309	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → ((𝐹:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽) ↔ ∀𝑦 ∈ (𝐾 ↾t 𝐵)(◡𝐹 “ 𝑦) ∈ 𝐽))
59	55	biantrurd 542	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → (∀𝑦 ∈ (𝐾 ↾t 𝐵)(◡𝐹 “ 𝑦) ∈ 𝐽 ↔ (𝐹:∪ 𝐽⟶𝐵 ∧ ∀𝑦 ∈ (𝐾 ↾t 𝐵)(◡𝐹 “ 𝑦) ∈ 𝐽)))
60	58, 59	bitrd 282	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → ((𝐹:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽) ↔ (𝐹:∪ 𝐽⟶𝐵 ∧ ∀𝑦 ∈ (𝐾 ↾t 𝐵)(◡𝐹 “ 𝑦) ∈ 𝐽)))
61		simprl 783	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐽 ∈ Top)
62	61, 15	sylib 221	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
63		iscn 23608	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
64	62, 28, 63	syl2anc 596	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
65	18	adantr 486	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
66		iscn 23608	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵)) → (𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵)) ↔ (𝐹:∪ 𝐽⟶𝐵 ∧ ∀𝑦 ∈ (𝐾 ↾t 𝐵)(◡𝐹 “ 𝑦) ∈ 𝐽)))
67	62, 65, 66	syl2anc 596	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → (𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵)) ↔ (𝐹:∪ 𝐽⟶𝐵 ∧ ∀𝑦 ∈ (𝐾 ↾t 𝐵)(◡𝐹 “ 𝑦) ∈ 𝐽)))
68	60, 64, 67	3bitr4d 314	. . 3 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))))
69	68	ex 418	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → ((𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵)))))
70	12, 24, 69	pm5.21ndd 382	1 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∪ cuni 4872  ^◡ccnv 5657  dom cdm 5658  ran crn 5659   “ cima 5661  Fun wfun 6560   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ↾_t crest 17641  Topctop 23266  TopOnctopon 23283   Cn ccn 23597

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-map 8887  df-en 9012  df-fin 9015  df-fi 9440  df-rest 17643  df-topgen 17664  df-top 23267  df-topon 23284  df-bases 23319  df-cn 23600

This theorem is used by:  cnrest2r  23660  rncmp  23769  connima  23798  conncn  23799  kgencn2  23931  kgencn3  23932  qtoprest  24091  hmeores  24145  efmndtmd  24475  submtmd  24478  subgtgp  24479  symgtgp  24480  metdcn2  25214  metdscn2  25232  cnmptre  25303  iimulcn  25314  icchmeo  25317  evth  25335  evth2  25336  lebnumlem2  25338  reparphti  25373  efrlim  27354  rmulccn  34671  raddcn  34672  xrge0mulc1cn  34684  cvxpconn  36215  cvxsconn  36216  cvmliftmolem1  36254  cvmliftlem8  36265  cvmlift2lem9  36284  cvmlift3lem6  36297  ivthALT  37338  knoppcnlem10  37583  broucube  38840  areacirclem2  38895  cnres2  38963  cnresima  38964  refsumcn  46321  icccncfext  47170