Theorem iscnp4 23150

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 23137	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1118	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1169	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
4		simplr 768	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
5		simpll2 1214	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝐾 ∈ (TopOn‘𝑌))
6		topontop 22800	. . . . . . . . 9 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
7	5, 6	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝐾 ∈ Top)
8		eqid 2729	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
9	8	neii1 22993	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝑦 ⊆ ∪ 𝐾)
10	7, 9	sylancom 588	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝑦 ⊆ ∪ 𝐾)
11	8	ntropn 22936	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐾) → ((int‘𝐾)‘𝑦) ∈ 𝐾)
12	7, 10, 11	syl2anc 584	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → ((int‘𝐾)‘𝑦) ∈ 𝐾)
13		simpr 484	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}))
14	3	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝐹:𝑋⟶𝑌)
15		simpll3 1215	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝑃 ∈ 𝑋)
16	14, 15	ffvelcdmd 7057	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → (𝐹‘𝑃) ∈ 𝑌)
17		toponuni 22801	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
18	5, 17	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝑌 = ∪ 𝐾)
19	16, 18	eleqtrd 2830	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → (𝐹‘𝑃) ∈ ∪ 𝐾)
20	19	snssd 4773	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → {(𝐹‘𝑃)} ⊆ ∪ 𝐾)
21	8	neiint 22991	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ {(𝐹‘𝑃)} ⊆ ∪ 𝐾 ∧ 𝑦 ⊆ ∪ 𝐾) → (𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}) ↔ {(𝐹‘𝑃)} ⊆ ((int‘𝐾)‘𝑦)))
22	7, 20, 10, 21	syl3anc 1373	. . . . . . . . 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)))
23	13, 22	mpbid 232	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → {(𝐹‘𝑃)} ⊆ ((int‘𝐾)‘𝑦))
24		fvex 6871	. . . . . . . . 9 ⊢ (𝐹‘𝑃) ∈ V
25	24	snss 4749	. . . . . . . 8 ⊢ ((𝐹‘𝑃) ∈ ((int‘𝐾)‘𝑦) ↔ {(𝐹‘𝑃)} ⊆ ((int‘𝐾)‘𝑦))
26	23, 25	sylibr 234	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → (𝐹‘𝑃) ∈ ((int‘𝐾)‘𝑦))
27		cnpimaex 23143	. . . . . . 7 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ ((int‘𝐾)‘𝑦) ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ ((int‘𝐾)‘𝑦)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ ((int‘𝐾)‘𝑦)))
28	4, 12, 26, 27	syl3anc 1373	. . . . . 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)))
29		simpl1 1192	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
30	29	ad2antrr 726	. . . . . . . 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))
31		topontop 22800	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
32	30, 31	syl 17	. . . . . . 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)
33		simprl 770	. . . . . . 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)
34		simprrl 780	. . . . . . 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)
35		opnneip 23006	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑃 ∈ 𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝑃}))
36	32, 33, 34, 35	syl3anc 1373	. . . . . 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}))
37		simprrr 781	. . . . . . 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))
38	8	ntrss2 22944	. . . . . . . . 9 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐾) → ((int‘𝐾)‘𝑦) ⊆ 𝑦)
39	7, 10, 38	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → ((int‘𝐾)‘𝑦) ⊆ 𝑦)
40	39	adantr 480	. . . . . . 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)
41	37, 40	sstrd 3957	. . . . . 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)
42	28, 36, 41	reximssdv 3151	. . . . 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)
43	42	ralrimiva 3125	. . . 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)
44	3, 43	jca 511	. . 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))
45	44	ex 412	. 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)))
46		simpll2 1214	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝐾 ∈ (TopOn‘𝑌))
47	46, 6	syl 17	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝐾 ∈ Top)
48		simprl 770	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝑦 ∈ 𝐾)
49		simprr 772	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → (𝐹‘𝑃) ∈ 𝑦)
50		opnneip 23006	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}))
51	47, 48, 49, 50	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}))
52		simpl1 1192	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
53	52	ad2antrr 726	. . . . . . . . . . . . 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))
54	53, 31	syl 17	. . . . . . . . . . . 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)
55		simprl 770	. . . . . . . . . . . . 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}))
56		eqid 2729	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
57	56	neii1 22993	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑥 ⊆ ∪ 𝐽)
58	54, 55, 57	syl2anc 584	. . . . . . . . . . . 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)
59	56	ntropn 22936	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
60	54, 58, 59	syl2anc 584	. . . . . . . . . . 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)
61		simpll3 1215	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝑃 ∈ 𝑋)
62	61	adantr 480	. . . . . . . . . . . . . . . 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)
63		toponuni 22801	. . . . . . . . . . . . . . . . 17 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
64	53, 63	syl 17	. . . . . . . . . . . . . . . 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)
65	62, 64	eleqtrd 2830	. . . . . . . . . . . . . . 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)
66	65	snssd 4773	. . . . . . . . . . . . . 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)
67	56	neiint 22991	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ ∪ 𝐽 ∧ 𝑥 ⊆ ∪ 𝐽) → (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
68	54, 66, 58, 67	syl3anc 1373	. . . . . . . . . . . . 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)))
69	55, 68	mpbid 232	. . . . . . . . . . . 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))
70		snssg 4747	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ 𝑋 → (𝑃 ∈ ((int‘𝐽)‘𝑥) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
71	62, 70	syl 17	. . . . . . . . . . . 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)))
72	69, 71	mpbird 257	. . . . . . . . . . 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))
73	56	ntrss2 22944	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
74	54, 58, 73	syl2anc 584	. . . . . . . . . . . . 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)
75		imass2 6073	. . . . . . . . . . . . 13 ⊢ (((int‘𝐽)‘𝑥) ⊆ 𝑥 → (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ (𝐹 “ 𝑥))
76	74, 75	syl 17	. . . . . . . . . . . 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))
77		simprr 772	. . . . . . . . . . . 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)
78	76, 77	sstrd 3957	. . . . . . . . . . 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)
79		eleq2 2817	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((int‘𝐽)‘𝑥) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ ((int‘𝐽)‘𝑥)))
80		imaeq2 6027	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((int‘𝐽)‘𝑥) → (𝐹 “ 𝑧) = (𝐹 “ ((int‘𝐽)‘𝑥)))
81	80	sseq1d 3978	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((int‘𝐽)‘𝑥) → ((𝐹 “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦))
82	79, 81	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑧 = ((int‘𝐽)‘𝑥) → ((𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ ((int‘𝐽)‘𝑥) ∧ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦)))
83	82	rspcev 3588	. . . . . . . . . . 11 ⊢ ((((int‘𝐽)‘𝑥) ∈ 𝐽 ∧ (𝑃 ∈ ((int‘𝐽)‘𝑥) ∧ (𝐹 “ ((int‘𝐽)‘𝑥)) ⊆ 𝑦)) → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦))
84	60, 72, 78, 83	syl12anc 836	. . . . . . . . . 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))
85	84	rexlimdvaa 3135	. . . . . . . . 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)))
86	51, 85	embantd 59	. . . . . . . 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)))
87	86	ex 412	. . . . . . 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))))
88	87	com23 86	. . . . . 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))))
89	88	exp4a 431	. . . . 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)))))
90	89	ralimdv2 3142	. . . 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))))
91	90	imdistanda 571	. . 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)))))
92		iscnp 23124	. . 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)))))
93	91, 92	sylibrd 259	. 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)))
94	45, 93	impbid 212	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   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914  {csn 4589  ∪ cuni 4871   “ cima 5641  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Topctop 22780  TopOnctopon 22797  intcnt 22904  neicnei 22984   CnP ccnp 23112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-top 22781  df-topon 22798  df-ntr 22907  df-nei 22985  df-cnp 23115

This theorem is referenced by:  cnnei  23169