Theorem iscnp4 14538

Description: The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃 " in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
iscnp4	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝑃,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Proof of Theorem iscnp4

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnpf2 14527	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1205	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1157	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
4		simpll1 1038	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝐽 ∈ (TopOn‘𝑋))
5		simpll2 1039	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝐾 ∈ (TopOn‘𝑌))
6		simpll3 1040	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝑃 ∈ 𝑋)
7		simplr 528	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
8		topontop 14334	. . . . . . . . 9 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
9	5, 8	syl 14	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝐾 ∈ Top)
10		eqid 2196	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
11	10	neii1 14467	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝑦 ⊆ ∪ 𝐾)
12	9, 11	sylancom 420	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝑦 ⊆ ∪ 𝐾)
13	10	ntropn 14437	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐾) → ((int‘𝐾)‘𝑦) ∈ 𝐾)
14	9, 12, 13	syl2anc 411	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → ((int‘𝐾)‘𝑦) ∈ 𝐾)
15		simpr 110	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}))
16	3	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝐹:𝑋⟶𝑌)
17	16, 6	ffvelcdmd 5701	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → (𝐹‘𝑃) ∈ 𝑌)
18		toponuni 14335	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
19	5, 18	syl 14	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝑌 = ∪ 𝐾)
20	17, 19	eleqtrd 2275	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → (𝐹‘𝑃) ∈ ∪ 𝐾)
21	20	snssd 3768	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → {(𝐹‘𝑃)} ⊆ ∪ 𝐾)
22	10	neiint 14465	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ {(𝐹‘𝑃)} ⊆ ∪ 𝐾 ∧ 𝑦 ⊆ ∪ 𝐾) → (𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}) ↔ {(𝐹‘𝑃)} ⊆ ((int‘𝐾)‘𝑦)))
23	9, 21, 12, 22	syl3anc 1249	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → (𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}) ↔ {(𝐹‘𝑃)} ⊆ ((int‘𝐾)‘𝑦)))
24	15, 23	mpbid 147	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → {(𝐹‘𝑃)} ⊆ ((int‘𝐾)‘𝑦))
25		fvexg 5580	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑃 ∈ 𝑋) → (𝐹‘𝑃) ∈ V)
26	7, 6, 25	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → (𝐹‘𝑃) ∈ V)
27		snssg 3757	. . . . . . . . 9 ⊢ ((𝐹‘𝑃) ∈ V → ((𝐹‘𝑃) ∈ ((int‘𝐾)‘𝑦) ↔ {(𝐹‘𝑃)} ⊆ ((int‘𝐾)‘𝑦)))
28	26, 27	syl 14	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → ((𝐹‘𝑃) ∈ ((int‘𝐾)‘𝑦) ↔ {(𝐹‘𝑃)} ⊆ ((int‘𝐾)‘𝑦)))
29	24, 28	mpbird 167	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → (𝐹‘𝑃) ∈ ((int‘𝐾)‘𝑦))
30		icnpimaex 14531	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ ((int‘𝐾)‘𝑦) ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ ((int‘𝐾)‘𝑦))) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦)))
31	4, 5, 6, 7, 14, 29, 30	syl33anc 1264	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦)))
32		simpl1 1002	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
33	32	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) ∧ (𝑥 ∈ 𝐽 ∧ (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → 𝐽 ∈ (TopOn‘𝑋))
34		topontop 14334	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
35	33, 34	syl 14	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) ∧ (𝑥 ∈ 𝐽 ∧ (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → 𝐽 ∈ Top)
36		simprl 529	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) ∧ (𝑥 ∈ 𝐽 ∧ (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → 𝑥 ∈ 𝐽)
37		simprrl 539	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) ∧ (𝑥 ∈ 𝐽 ∧ (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → 𝑃 ∈ 𝑥)
38		opnneip 14479	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑃 ∈ 𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝑃}))
39	35, 36, 37, 38	syl3anc 1249	. . . . . 6 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) ∧ (𝑥 ∈ 𝐽 ∧ (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → 𝑥 ∈ ((nei‘𝐽)‘{𝑃}))
40		simprrr 540	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) ∧ (𝑥 ∈ 𝐽 ∧ (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦))
41	10	ntrss2 14441	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐾) → ((int‘𝐾)‘𝑦) ⊆ 𝑦)
42	9, 12, 41	syl2anc 411	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → ((int‘𝐾)‘𝑦) ⊆ 𝑦)
43	42	adantr 276	. . . . . . 7 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) ∧ (𝑥 ∈ 𝐽 ∧ (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → ((int‘𝐾)‘𝑦) ⊆ 𝑦)
44	40, 43	sstrd 3194	. . . . . 6 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) ∧ (𝑥 ∈ 𝐽 ∧ (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦)))) → (𝐹 “ 𝑥) ⊆ 𝑦)
45	31, 39, 44	reximssdv 2601	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → ∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦)
46	45	ralrimiva 2570	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦)
47	3, 46	jca 306	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦))
48	47	ex 115	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦)))
49		simpll2 1039	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝐾 ∈ (TopOn‘𝑌))
50	49, 8	syl 14	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝐾 ∈ Top)
51		simprl 529	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝑦 ∈ 𝐾)
52		simprr 531	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → (𝐹‘𝑃) ∈ 𝑦)
53		opnneip 14479	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}))
54	50, 51, 52, 53	syl3anc 1249	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}))
55		simpl1 1002	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
56	55	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → 𝐽 ∈ (TopOn‘𝑋))
57	56, 34	syl 14	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → 𝐽 ∈ Top)
58		simprl 529	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → 𝑥 ∈ ((nei‘𝐽)‘{𝑃}))
59		eqid 2196	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
60	59	neii1 14467	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑥 ⊆ ∪ 𝐽)
61	57, 58, 60	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → 𝑥 ⊆ ∪ 𝐽)
62	59	ntropn 14437	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
63	57, 61, 62	syl2anc 411	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
64		simpll3 1040	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝑃 ∈ 𝑋)
65	64	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → 𝑃 ∈ 𝑋)
66		toponuni 14335	. . . . . . . . . . . . . . . . 17 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
67	56, 66	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → 𝑋 = ∪ 𝐽)
68	65, 67	eleqtrd 2275	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → 𝑃 ∈ ∪ 𝐽)
69	68	snssd 3768	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → {𝑃} ⊆ ∪ 𝐽)
70	59	neiint 14465	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ ∪ 𝐽 ∧ 𝑥 ⊆ ∪ 𝐽) → (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
71	57, 69, 61, 70	syl3anc 1249	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
72	58, 71	mpbid 147	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → {𝑃} ⊆ ((int‘𝐽)‘𝑥))
73		snssg 3757	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ 𝑋 → (𝑃 ∈ ((int‘𝐽)‘𝑥) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
74	65, 73	syl 14	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → (𝑃 ∈ ((int‘𝐽)‘𝑥) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
75	72, 74	mpbird 167	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → 𝑃 ∈ ((int‘𝐽)‘𝑥))
76	59	ntrss2 14441	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
77	57, 61, 76	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
78		imass2 5046	. . . . . . . . . . . . 13 ⊢ (((int‘𝐽)‘𝑥) ⊆ 𝑥 → (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ (𝐹 “ 𝑥))
79	77, 78	syl 14	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ (𝐹 “ 𝑥))
80		simprr 531	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → (𝐹 “ 𝑥) ⊆ 𝑦)
81	79, 80	sstrd 3194	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦)
82		eleq2 2260	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((int‘𝐽)‘𝑥) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ ((int‘𝐽)‘𝑥)))
83		imaeq2 5006	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((int‘𝐽)‘𝑥) → (𝐹 “ 𝑧) = (𝐹 “ ((int‘𝐽)‘𝑥)))
84	83	sseq1d 3213	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((int‘𝐽)‘𝑥) → ((𝐹 “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦))
85	82, 84	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑧 = ((int‘𝐽)‘𝑥) → ((𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ ((int‘𝐽)‘𝑥) ∧ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦)))
86	85	rspcev 2868	. . . . . . . . . . 11 ⊢ ((((int‘𝐽)‘𝑥) ∈ 𝐽 ∧ (𝑃 ∈ ((int‘𝐽)‘𝑥) ∧ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦)) → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦))
87	63, 75, 81, 86	syl12anc 1247	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) ∧ (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦))
88	87	rexlimdvaa 2615	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → (∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦 → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦)))
89	54, 88	embantd 56	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → ((𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}) → ∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦)))
90	89	ex 115	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦) → ((𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}) → ∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦))))
91	90	com23 78	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}) → ∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦) → ((𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦) → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦))))
92	91	exp4a 366	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}) → ∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦) → (𝑦 ∈ 𝐾 → ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦)))))
93	92	ralimdv2 2567	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦 → ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦))))
94	93	imdistanda 448	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦)))))
95		iscnp 14519	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦)))))
96	94, 95	sylibrd 169	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)))
97	48, 96	impbid 129	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  Vcvv 2763   ⊆ wss 3157  {csn 3623  ∪ cuni 3840   “ cima 4667  ⟶wf 5255  ‘cfv 5259  (class class class)co 5925  Topctop 14317  TopOnctopon 14330  intcnt 14413  neicnei 14458   CnP ccnp 14506

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-top 14318  df-topon 14331  df-ntr 14416  df-nei 14459  df-cnp 14509

This theorem is referenced by:  cnnei  14552