Theorem cncnp 23642

Description: A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 15-May-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
cncnp	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))

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

Proof of Theorem cncnp

Dummy variables 𝑢 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscn 23597	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
2	1	simprbda 504	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
3		eqid 2763	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	cncnpi 23640	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
5	4	ralrimiva 3158	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ ∪ 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
6	5	adantl 487	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ ∪ 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
7		toponuni 23276	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
8	7	ad2antrr 739	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑋 = ∪ 𝐽)
9	6, 8	raleqtrrdv 3327	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
10	2, 9	jca 521	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
11		simprl 783	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹:𝑋⟶𝑌)
12		cnvimass 6215	. . . . . . . . . 10 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
13		fdm 6747	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
14	13	adantl 487	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) → dom 𝐹 = 𝑋)
15	12, 14	sseqtrid 3955	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) → (◡𝐹 “ 𝑦) ⊆ 𝑋)
16		ssralv 3982	. . . . . . . . 9 ⊢ ((◡𝐹 “ 𝑦) ⊆ 𝑋 → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (◡𝐹 “ 𝑦)𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
17	15, 16	syl 18	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (◡𝐹 “ 𝑦)𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
18		simprr 785	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
19		simpllr 788	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑦 ∈ 𝐾)
20		ffn 6737	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶𝑌 → 𝐹 Fn 𝑋)
21	20	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 Fn 𝑋)
22		simprl 783	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑥 ∈ (◡𝐹 “ 𝑦))
23		elpreima 7086	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝑋 → (𝑥 ∈ (◡𝐹 “ 𝑦) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ 𝑦)))
24	23	simplbda 505	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ (◡𝐹 “ 𝑦)) → (𝐹‘𝑥) ∈ 𝑦)
25	21, 22, 24	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (𝐹‘𝑥) ∈ 𝑦)
26		cnpimaex 23618	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦))
27	18, 19, 25, 26	syl3anc 1399	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦))
28		simpllr 788	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢 ∈ 𝐽) → 𝐹:𝑋⟶𝑌)
29	28	ffund 6742	. . . . . . . . . . . . . 14 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢 ∈ 𝐽) → Fun 𝐹)
30		simp-4l 795	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐽 ∈ (TopOn‘𝑋))
31		toponss 23289	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ 𝑋)
32	30, 31	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ 𝑋)
33	28, 13	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢 ∈ 𝐽) → dom 𝐹 = 𝑋)
34	32, 33	sseqtrrd 3950	. . . . . . . . . . . . . 14 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ dom 𝐹)
35		funimass3 7082	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ 𝑢 ⊆ dom 𝐹) → ((𝐹 “ 𝑢) ⊆ 𝑦 ↔ 𝑢 ⊆ (◡𝐹 “ 𝑦)))
36	29, 34, 35	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢 ∈ 𝐽) → ((𝐹 “ 𝑢) ⊆ 𝑦 ↔ 𝑢 ⊆ (◡𝐹 “ 𝑦)))
37	36	anbi2d 642	. . . . . . . . . . . 12 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢 ∈ 𝐽) → ((𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦) ↔ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ 𝑦))))
38	37	rexbidva 3188	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦) ↔ ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ 𝑦))))
39	27, 38	mpbid 235	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ 𝑦)))
40	39	expr 462	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ (◡𝐹 “ 𝑦)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ 𝑦))))
41	40	ralimdva 3178	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ (◡𝐹 “ 𝑦)𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (◡𝐹 “ 𝑦)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ 𝑦))))
42	17, 41	syld 48	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (◡𝐹 “ 𝑦)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ 𝑦))))
43	42	impr 460	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∀𝑥 ∈ (◡𝐹 “ 𝑦)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ 𝑦)))
44	43	an32s 665	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦 ∈ 𝐾) → ∀𝑥 ∈ (◡𝐹 “ 𝑦)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ 𝑦)))
45		topontop 23275	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
46	45	ad3antrrr 743	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦 ∈ 𝐾) → 𝐽 ∈ Top)
47		eltop2 23337	. . . . . 6 ⊢ (𝐽 ∈ Top → ((◡𝐹 “ 𝑦) ∈ 𝐽 ↔ ∀𝑥 ∈ (◡𝐹 “ 𝑦)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ 𝑦))))
48	46, 47	syl 18	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦 ∈ 𝐾) → ((◡𝐹 “ 𝑦) ∈ 𝐽 ↔ ∀𝑥 ∈ (◡𝐹 “ 𝑦)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ 𝑦))))
49	44, 48	mpbird 260	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
50	49	ralrimiva 3158	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)
51	1	adantr 486	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
52	11, 50, 51	mpbir2and 726	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾))
53	10, 52	impbida 813	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ∪ cuni 4870  ^◡ccnv 5655  dom cdm 5656   “ cima 5659  Fun wfun 6558   Fn wfn 6559  ⟶wf 6560  ‘cfv 6564  (class class class)co 7449  Topctop 23255  TopOnctopon 23272   Cn ccn 23586   CnP ccnp 23587

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-fv 6572  df-ov 7452  df-oprab 7453  df-mpo 7454  df-1st 8033  df-2nd 8034  df-map 8885  df-topgen 17655  df-top 23256  df-topon 23273  df-cn 23589  df-cnp 23590

This theorem is used by:  cncnp2  23643  cnnei  23644  cnconst2  23645  1stccn  23826  ptcn  23990  cnflf  24365  cnfcf  24405  symgtgp  24469  ghmcnp  24478  metcn  24906  txmetcn  24911  cnlimc  26252  dvcn  26285  dvcnvre  26383  psercn  26799  abelth  26814  cxpcn3  27122  cvmlift2lem11  36279  cvmlift2lem12  36280  cvmlift3lem8  36292  ioccncflimc  47162  cncfuni  47163  icccncfext  47164  icocncflimc  47166  cncfiooicclem1  47170  dirkercncflem2  47381  dirkercncflem4  47383  dirkercncf  47384  fourierdlem32  47416  fourierdlem33  47417  fourierdlem62  47445  fourierdlem93  47476  fourierdlem101  47484