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Theorem cnrest2 21888

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 21842	. . . 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 2821	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2821	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 21848	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	5	ffnd 6509	. . . . . 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 1133	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → ran 𝐹 ⊆ 𝐵)
9	7, 8	jctird 529	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 ⊆ 𝐵)))
10		df-f 6353	. . . 4 ⊢ (𝐹:∪ 𝐽⟶𝐵 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 ⊆ 𝐵))
11	9, 10	syl6ibr 254	. . 3 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝐵))
12	2, 11	jcad 515	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)))
13		cntop1 21842	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn (𝐾 ↾_t 𝐵)) → 𝐽 ∈ Top)
14	13	adantl 484	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾_t 𝐵))) → 𝐽 ∈ Top)
15		toptopon2 21520	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
16	14, 15	sylib 220	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾_t 𝐵))) → 𝐽 ∈ (TopOn‘∪ 𝐽))
17		resttopon 21763	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ⊆ 𝑌) → (𝐾 ↾_t 𝐵) ∈ (TopOn‘𝐵))
18	17	3adant2 1127	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐾 ↾_t 𝐵) ∈ (TopOn‘𝐵))
19	18	adantr 483	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾_t 𝐵))) → (𝐾 ↾_t 𝐵) ∈ (TopOn‘𝐵))
20		simpr 487	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾_t 𝐵))) → 𝐹 ∈ (𝐽 Cn (𝐾 ↾_t 𝐵)))
21		cnf2 21851	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (𝐾 ↾_t 𝐵) ∈ (TopOn‘𝐵) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾_t 𝐵))) → 𝐹:∪ 𝐽⟶𝐵)
22	16, 19, 20, 21	syl3anc 1367	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾_t 𝐵))) → 𝐹:∪ 𝐽⟶𝐵)
23	14, 22	jca 514	. . 3 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾_t 𝐵))) → (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵))
24	23	ex 415	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn (𝐾 ↾_t 𝐵)) → (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)))
25		vex 3497	. . . . . . . . 9 ⊢ 𝑥 ∈ V
26	25	inex1 5213	. . . . . . . 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 1187	. . . . . . . 8 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐾 ∈ (TopOn‘𝑌))
29		toponmax 21528	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
30	28, 29	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝑌 ∈ 𝐾)
31		simpl3 1189	. . . . . . . . 9 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐵 ⊆ 𝑌)
32	30, 31	ssexd 5220	. . . . . . . 8 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐵 ∈ V)
33		elrest 16695	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ V) → (𝑦 ∈ (𝐾 ↾_t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑦 = (𝑥 ∩ 𝐵)))
34	28, 32, 33	syl2anc 586	. . . . . . 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 5919	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → (^◡𝐹 “ 𝑦) = (^◡𝐹 “ (𝑥 ∩ 𝐵)))
36	35	eleq1d 2897	. . . . . . . 8 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((^◡𝐹 “ 𝑦) ∈ 𝐽 ↔ (^◡𝐹 “ (𝑥 ∩ 𝐵)) ∈ 𝐽))
37	36	adantl 484	. . . . . . 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 5302	. . . . . 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 776	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → 𝐹:∪ 𝐽⟶𝐵)
40		ffun 6511	. . . . . . . . . 10 ⊢ (𝐹:∪ 𝐽⟶𝐵 → Fun 𝐹)
41		inpreima 6828	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (^◡𝐹 “ (𝑥 ∩ 𝐵)) = ((^◡𝐹 “ 𝑥) ∩ (^◡𝐹 “ 𝐵)))
42	39, 40, 41	3syl 18	. . . . . . . . 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 5943	. . . . . . . . . . . 12 ⊢ (^◡𝐹 “ 𝑥) ⊆ dom 𝐹
44		cnvimarndm 5944	. . . . . . . . . . . 12 ⊢ (^◡𝐹 “ ran 𝐹) = dom 𝐹
45	43, 44	sseqtrri 4003	. . . . . . . . . . 11 ⊢ (^◡𝐹 “ 𝑥) ⊆ (^◡𝐹 “ ran 𝐹)
46		simpll2 1209	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → ran 𝐹 ⊆ 𝐵)
47		imass2 5959	. . . . . . . . . . . 12 ⊢ (ran 𝐹 ⊆ 𝐵 → (^◡𝐹 “ ran 𝐹) ⊆ (^◡𝐹 “ 𝐵))
48	46, 47	syl 17	. . . . . . . . . . 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 3977	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → (^◡𝐹 “ 𝑥) ⊆ (^◡𝐹 “ 𝐵))
50		df-ss 3951	. . . . . . . . . 10 ⊢ ((^◡𝐹 “ 𝑥) ⊆ (^◡𝐹 “ 𝐵) ↔ ((^◡𝐹 “ 𝑥) ∩ (^◡𝐹 “ 𝐵)) = (^◡𝐹 “ 𝑥))
51	49, 50	sylib 220	. . . . . . . . 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 2856	. . . . . . . 8 ⊢ ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) ∧ 𝑥 ∈ 𝐾) → (^◡𝐹 “ (𝑥 ∩ 𝐵)) = (^◡𝐹 “ 𝑥))
53	52	eleq1d 2897	. . . . . . 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 3196	. . . . . 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 771	. . . . . . . 8 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐹:∪ 𝐽⟶𝐵)
56	55, 31	fssd 6522	. . . . . . 7 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐹:∪ 𝐽⟶𝑌)
57	56	biantrurd 535	. . . . . 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 308	. . . . 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 535	. . . . 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 281	. . . 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 769	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐽 ∈ Top)
62	61, 15	sylib 220	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
63		iscn 21837	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (^◡𝐹 “ 𝑥) ∈ 𝐽)))
64	62, 28, 63	syl2anc 586	. . . 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 483	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹:∪ 𝐽⟶𝐵)) → (𝐾 ↾_t 𝐵) ∈ (TopOn‘𝐵))
66		iscn 21837	. . . . 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 586	. . . 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 313	. . 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 415	. 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 383	1 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾_t 𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ∪ cuni 4831 ^◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 Fun wfun 6343 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↾_t crest 16688 Topctop 21495 TopOnctopon 21512 Cn ccn 21826

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-fin 8507 df-fi 8869 df-rest 16690 df-topgen 16711 df-top 21496 df-topon 21513 df-bases 21548 df-cn 21829

This theorem is referenced by: cnrest2r 21889 rncmp 21998 connima 22027 conncn 22028 kgencn2 22159 kgencn3 22160 qtoprest 22319 hmeores 22373 efmndtmd 22703 submtmd 22706 subgtgp 22707 symgtgp 22708 metdcn2 23441 metdscn2 23459 cnmptre 23525 iimulcn 23536 icchmeo 23539 evth 23557 evth2 23558 lebnumlem2 23560 reparphti 23595 efrlim 25541 rmulccn 31166 raddcn 31167 xrge0mulc1cn 31179 cvxpconn 32484 cvxsconn 32485 cvmliftmolem1 32523 cvmliftlem8 32534 cvmlift2lem9 32553 cvmlift3lem6 32566 ivthALT 33678 knoppcnlem10 33836 broucube 34920 areacirclem2 34977 cnres2 35035 cnresima 35036 refsumcn 41280 icccncfext 42163

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