Theorem cnrest2 23412

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 23366	. . . 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 2769	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2769	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 23372	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	5	ffnd 6707	. . . . . 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 1153	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → ran 𝐹 ⊆ 𝐵)
9	7, 8	jctird 535	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 ⊆ 𝐵)))
10		df-f 6541	. . . 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 521	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)))
13		cntop1 23366	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵)) → 𝐽 ∈ Top)
14	13	adantl 486	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → 𝐽 ∈ Top)
15		toptopon2 23044	. . . . . 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 23287	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ⊆ 𝑌) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
18	17	3adant2 1147	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
19	18	adantr 485	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
20		simpr 489	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵)))
21		cnf2 23375	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → 𝐹:∪ 𝐽⟶𝐵)
22	16, 19, 20, 21	syl3anc 1396	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → 𝐹:∪ 𝐽⟶𝐵)
23	14, 22	jca 520	. . 3 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵))) → (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵))
24	23	ex 417	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn (𝐾 ↾t 𝐵)) → (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)))
25		vex 3467	. . . . . . . . 9 ⊢ 𝑥 ∈ V
26	25	inex1 5288	. . . . . . . 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 1208	. . . . . . . 8 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐾 ∈ (TopOn‘𝑌))
29		toponmax 23052	. . . . . . . . . 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 1210	. . . . . . . . 9 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐵 ⊆ 𝑌)
32	30, 31	ssexd 5295	. . . . . . . 8 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐵 ∈ V)
33		elrest 17480	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ V) → (𝑦 ∈ (𝐾 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑦 = (𝑥 ∩ 𝐵)))
34	28, 32, 33	syl2anc 595	. . . . . . 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 6059	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (𝑥 ∩ 𝐵)))
36	35	eleq1d 2854	. . . . . . . 8 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((◡𝐹 “ 𝑦) ∈ 𝐽 ↔ (◡𝐹 “ (𝑥 ∩ 𝐵)) ∈ 𝐽))
37	36	adantl 486	. . . . . . 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 5382	. . . . . 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 789	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → 𝐹:∪ 𝐽⟶𝐵)
40		ffun 6709	. . . . . . . . . 10 ⊢ (𝐹:∪ 𝐽⟶𝐵 → Fun 𝐹)
41		inpreima 7060	. . . . . . . . . 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 6085	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
44		cnvimarndm 6086	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
45	43, 44	sseqtrri 3994	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ ran 𝐹)
46		simpll2 1230	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → ran 𝐹 ⊆ 𝐵)
47		imass2 6105	. . . . . . . . . . . 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 3956	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝐵))
50		dfss2 3931	. . . . . . . . . 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 2804	. . . . . . . 8 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ (𝑥 ∩ 𝐵)) = (◡𝐹 “ 𝑥))
53	52	eleq1d 2854	. . . . . . 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 3192	. . . . . 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 784	. . . . . . . 8 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐹:∪ 𝐽⟶𝐵)
56	55, 31	fssd 6724	. . . . . . 7 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐹:∪ 𝐽⟶𝑌)
57	56	biantrurd 541	. . . . . 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 541	. . . . 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 782	. . . . . 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 23361	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)))
64	62, 28, 63	syl2anc 595	. . . 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 485	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵))
66		iscn 23361	. . . . 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 595	. . . 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 417	. 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))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ∪ cuni 4876  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   “ cima 5665  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ↾_t crest 17473  Topctop 23019  TopOnctopon 23036   Cn ccn 23350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-map 8826  df-en 8944  df-fin 8947  df-fi 9371  df-rest 17475  df-topgen 17496  df-top 23020  df-topon 23037  df-bases 23072  df-cn 23353

This theorem is referenced by:  cnrest2r  23413  rncmp  23522  connima  23551  conncn  23552  kgencn2  23683  kgencn3  23684  qtoprest  23843  hmeores  23897  efmndtmd  24227  submtmd  24230  subgtgp  24231  symgtgp  24232  metdcn2  24966  metdscn2  24984  cnmptre  25055  iimulcn  25066  icchmeo  25069  evth  25087  evth2  25088  lebnumlem2  25090  reparphti  25125  efrlim  27100  rmulccn  34263  raddcn  34264  xrge0mulc1cn  34276  cvxpconn  35633  cvxsconn  35634  cvmliftmolem1  35672  cvmliftlem8  35683  cvmlift2lem9  35702  cvmlift3lem6  35715  ivthALT  36735  knoppcnlem10  36980  broucube  38193  areacirclem2  38248  cnres2  38302  cnresima  38303  refsumcn  45642  icccncfext  46493