Theorem cncnp 23505

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 23460	. . . 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 2760	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	cncnpi 23503	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
5	4	ralrimiva 3154	. . . . 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 23139	. . . . 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 3323	. . 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 6078	. . . . . . . . . 10 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
13		fdm 6712	. . . . . . . . . . 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 3973	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) → (◡𝐹 “ 𝑦) ⊆ 𝑋)
16		ssralv 4000	. . . . . . . . 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 6702	. . . . . . . . . . . . . 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 7050	. . . . . . . . . . . . . 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 23481	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦))
27	18, 19, 25, 26	syl3anc 1398	. . . . . . . . . . 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 6707	. . . . . . . . . . . . . 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 23152	. . . . . . . . . . . . . . . 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 3968	. . . . . . . . . . . . . 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 7046	. . . . . . . . . . . . . 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 3184	. . . . . . . . . . 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 3174	. . . . . . . 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 23138	. . . . . . 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 23200	. . . . . 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 3154	. . 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 2145  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899  ∪ cuni 4867  ^◡ccnv 5654  dom cdm 5655   “ cima 5658  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  Topctop 23118  TopOnctopon 23135   Cn ccn 23449   CnP ccnp 23450

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-map 8828  df-topgen 17528  df-top 23119  df-topon 23136  df-cn 23452  df-cnp 23453

This theorem is used by:  cncnp2  23506  cnnei  23507  cnconst2  23508  1stccn  23689  ptcn  23853  cnflf  24228  cnfcf  24268  symgtgp  24332  ghmcnp  24341  metcn  24769  txmetcn  24774  cnlimc  26115  dvcn  26148  dvcnvre  26246  psercn  26662  abelth  26677  cxpcn3  26985  cvmlift2lem11  35892  cvmlift2lem12  35893  cvmlift3lem8  35905  ioccncflimc  46713  cncfuni  46714  icccncfext  46715  icocncflimc  46717  cncfiooicclem1  46721  dirkercncflem2  46932  dirkercncflem4  46934  dirkercncf  46935  fourierdlem32  46967  fourierdlem33  46968  fourierdlem62  46996  fourierdlem93  47027  fourierdlem101  47035