Theorem iscnp4 23287

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 23274	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1117	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1167	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
4		simplr 769	. . . . . . 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 1212	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → 𝐾 ∈ (TopOn‘𝑌))
6		topontop 22935	. . . . . . . . 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 2735	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
9	8	neii1 23130	. . . . . . . . 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 23073	. . . . . . . 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 1213	. . . . . . . . . . . . 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 7105	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})) → (𝐹‘𝑃) ∈ 𝑌)
17		toponuni 22936	. . . . . . . . . . . . 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 2841	. . . . . . . . . . 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 4814	. . . . . . . . . 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 23128	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ {(𝐹‘𝑃)} ⊆ ∪ 𝐾 ∧ 𝑦 ⊆ ∪ 𝐾) → (𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}) ↔ {(𝐹‘𝑃)} ⊆ ((int‘𝐾)‘𝑦)))
22	7, 20, 10, 21	syl3anc 1370	. . . . . . . . 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 6920	. . . . . . . . 9 ⊢ (𝐹‘𝑃) ∈ V
25	24	snss 4790	. . . . . . . 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 23280	. . . . . . 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 1370	. . . . . 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 1190	. . . . . . . . 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 22935	. . . . . . . 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 771	. . . . . . 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 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)))) → 𝑃 ∈ 𝑥)
35		opnneip 23143	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑃 ∈ 𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝑃}))
36	32, 33, 34, 35	syl3anc 1370	. . . . . 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 782	. . . . . . 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 23081	. . . . . . . . 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 4006	. . . . . 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 3171	. . . . 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 3144	. . . 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 1212	. . . . . . . . . . 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 771	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝑦 ∈ 𝐾)
49		simprr 773	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → (𝐹‘𝑃) ∈ 𝑦)
50		opnneip 23143	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}))
51	47, 48, 49, 50	syl3anc 1370	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)}))
52		simpl1 1190	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
57	56	neii1 23130	. . . . . . . . . . . . 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 23073	. . . . . . . . . . . 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 1213	. . . . . . . . . . . . . . . . 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 22936	. . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . 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 4814	. . . . . . . . . . . . . 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 23128	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ ∪ 𝐽 ∧ 𝑥 ⊆ ∪ 𝐽) → (𝑥 ∈ ((nei‘𝐽)‘{𝑃}) ↔ {𝑃} ⊆ ((int‘𝐽)‘𝑥)))
68	54, 66, 58, 67	syl3anc 1370	. . . . . . . . . . . . 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 4788	. . . . . . . . . . . . 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 23081	. . . . . . . . . . . . . 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 6123	. . . . . . . . . . . . 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 773	. . . . . . . . . . . 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 4006	. . . . . . . . . . 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 2828	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((int‘𝐽)‘𝑥) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ ((int‘𝐽)‘𝑥)))
80		imaeq2 6076	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((int‘𝐽)‘𝑥) → (𝐹 “ 𝑧) = (𝐹 “ ((int‘𝐽)‘𝑥)))
81	80	sseq1d 4027	. . . . . . . . . . . . 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 3622	. . . . . . . . . . 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 837	. . . . . . . . . 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 3154	. . . . . . . . 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 3161	. . . 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 23261	. . 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 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  {csn 4631  ∪ cuni 4912   “ cima 5692  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Topctop 22915  TopOnctopon 22932  intcnt 23041  neicnei 23121   CnP ccnp 23249

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-top 22916  df-topon 22933  df-ntr 23044  df-nei 23122  df-cnp 23252

This theorem is referenced by:  cnnei  23306