Theorem cncnp 21604

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 21559	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
2	1	simprbda 491	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
3		eqid 2772	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	cncnpi 21602	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
5	4	ralrimiva 3126	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ ∪ 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
6	5	adantl 474	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ ∪ 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
7		toponuni 21238	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
8	7	ad2antrr 713	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑋 = ∪ 𝐽)
9	8	raleqdv 3349	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑥 ∈ ∪ 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
10	6, 9	mpbird 249	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
11	2, 10	jca 504	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
12		simprl 758	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹:𝑋⟶𝑌)
13		cnvimass 5786	. . . . . . . . . 10 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
14		fdm 6349	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
15	14	adantl 474	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) → dom 𝐹 = 𝑋)
16	13, 15	syl5sseq 3903	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) → (◡𝐹 “ 𝑦) ⊆ 𝑋)
17		ssralv 3917	. . . . . . . . 9 ⊢ ((◡𝐹 “ 𝑦) ⊆ 𝑋 → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (◡𝐹 “ 𝑦)𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
18	16, 17	syl 17	. . . . . . . 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)))
19		simprr 760	. . . . . . . . . . . 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))
20		simpllr 763	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑦 ∈ 𝐾)
21		ffn 6341	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶𝑌 → 𝐹 Fn 𝑋)
22	21	ad2antlr 714	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 Fn 𝑋)
23		simprl 758	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑥 ∈ (◡𝐹 “ 𝑦))
24		elpreima 6651	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝑋 → (𝑥 ∈ (◡𝐹 “ 𝑦) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ 𝑦)))
25	24	simplbda 492	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ (◡𝐹 “ 𝑦)) → (𝐹‘𝑥) ∈ 𝑦)
26	22, 23, 25	syl2anc 576	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (𝐹‘𝑥) ∈ 𝑦)
27		cnpimaex 21580	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦))
28	19, 20, 26, 27	syl3anc 1351	. . . . . . . . . . 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))
29		simpllr 763	. . . . . . . . . . . . . . 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)
30	29	ffund 6345	. . . . . . . . . . . . . 14 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢 ∈ 𝐽) → Fun 𝐹)
31		simp-4l 770	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐽 ∈ (TopOn‘𝑋))
32		toponss 21251	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ 𝑋)
33	31, 32	sylan 572	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ 𝑋)
34	29, 14	syl 17	. . . . . . . . . . . . . . 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)
35	33, 34	sseqtr4d 3892	. . . . . . . . . . . . . 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)
36		funimass3 6647	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ 𝑢 ⊆ dom 𝐹) → ((𝐹 “ 𝑢) ⊆ 𝑦 ↔ 𝑢 ⊆ (◡𝐹 “ 𝑦)))
37	30, 35, 36	syl2anc 576	. . . . . . . . . . . . 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)))
38	37	anbi2d 619	. . . . . . . . . . . 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))))
39	38	rexbidva 3235	. . . . . . . . . . 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))))
40	28, 39	mpbid 224	. . . . . . . . . 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)))
41	40	expr 449	. . . . . . . . 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))))
42	41	ralimdva 3121	. . . . . . . 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))))
43	18, 42	syld 47	. . . . . . 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))))
44	43	impr 447	. . . . . 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)))
45	44	an32s 639	. . . . 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)))
46		topontop 21237	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
47	46	ad3antrrr 717	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦 ∈ 𝐾) → 𝐽 ∈ Top)
48		eltop2 21299	. . . . . 6 ⊢ (𝐽 ∈ Top → ((◡𝐹 “ 𝑦) ∈ 𝐽 ↔ ∀𝑥 ∈ (◡𝐹 “ 𝑦)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ 𝑦))))
49	47, 48	syl 17	. . . . 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))))
50	45, 49	mpbird 249	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
51	50	ralrimiva 3126	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)
52	1	adantr 473	. . 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)))
53	12, 51, 52	mpbir2and 700	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾))
54	11, 53	impbida 788	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∀wral 3082  ∃wrex 3083   ⊆ wss 3823  ∪ cuni 4708  ^◡ccnv 5402  dom cdm 5403   “ cima 5406  Fun wfun 6179   Fn wfn 6180  ⟶wf 6181  ‘cfv 6185  (class class class)co 6974  Topctop 21217  TopOnctopon 21234   Cn ccn 21548   CnP ccnp 21549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7499  df-2nd 7500  df-map 8206  df-topgen 16571  df-top 21218  df-topon 21235  df-cn 21551  df-cnp 21552

This theorem is referenced by:  cncnp2  21605  cnnei  21606  cnconst2  21607  1stccn  21787  ptcn  21951  cnflf  22326  cnfcf  22366  symgtgp  22425  ghmcnp  22438  metcn  22868  txmetcn  22873  cnlimc  24201  dvcn  24233  dvcnvre  24331  psercn  24729  abelth  24744  cxpcn3  25042  cvmlift2lem11  32174  cvmlift2lem12  32175  cvmlift3lem8  32187  ioccncflimc  41623  cncfuni  41624  icccncfext  41625  icocncflimc  41627  cncfiooicclem1  41631  dirkercncflem2  41845  dirkercncflem4  41847  dirkercncf  41848  fourierdlem32  41880  fourierdlem33  41881  fourierdlem62  41909  fourierdlem93  41940  fourierdlem101  41948