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Theorem cncnp 14182

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 14149	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (^◡𝐹 “ 𝑦) ∈ 𝐽)))
2	1	simprbda 383	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌)
3		eqid 2189	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	cncnpi 14180	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ ∪ 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
5	4	ralrimiva 2563	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ ∪ 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
6	5	adantl 277	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ ∪ 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
7		toponuni 13967	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
8	7	ad2antrr 488	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑋 = ∪ 𝐽)
9	8	raleqdv 2692	. . . 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 167	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
11	2, 10	jca 306	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
12		simprl 529	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹:𝑋⟶𝑌)
13		cnvimass 5009	. . . . . . . . . 10 ⊢ (^◡𝐹 “ 𝑦) ⊆ dom 𝐹
14		fdm 5390	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
15	14	adantl 277	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) → dom 𝐹 = 𝑋)
16	13, 15	sseqtrid 3220	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) → (^◡𝐹 “ 𝑦) ⊆ 𝑋)
17		ssralv 3234	. . . . . . . . 9 ⊢ ((^◡𝐹 “ 𝑦) ⊆ 𝑋 → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (^◡𝐹 “ 𝑦)𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
18	16, 17	syl 14	. . . . . . . 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		simp-4l 541	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (^◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐽 ∈ (TopOn‘𝑋))
20		simp-4r 542	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (^◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐾 ∈ (TopOn‘𝑌))
21		topontop 13966	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
22	20, 21	syl 14	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (^◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐾 ∈ Top)
23		simprr 531	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (^◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
24		cnprcl2k 14158	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → 𝑥 ∈ 𝑋)
25	19, 22, 23, 24	syl3anc 1249	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (^◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑥 ∈ 𝑋)
26		simpllr 534	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (^◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑦 ∈ 𝐾)
27		ffn 5384	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶𝑌 → 𝐹 Fn 𝑋)
28	27	ad2antlr 489	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (^◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 Fn 𝑋)
29		simprl 529	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (^◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑥 ∈ (^◡𝐹 “ 𝑦))
30		elpreima 5655	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝑋 → (𝑥 ∈ (^◡𝐹 “ 𝑦) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) ∈ 𝑦)))
31	30	simplbda 384	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ (^◡𝐹 “ 𝑦)) → (𝐹‘𝑥) ∈ 𝑦)
32	28, 29, 31	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (^◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (𝐹‘𝑥) ∈ 𝑦)
33		icnpimaex 14163	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦))
34	19, 20, 25, 23, 26, 32, 33	syl33anc 1264	. . . . . . . . . . 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))
35		simpllr 534	. . . . . . . . . . . . . . 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)
36	35	ffund 5388	. . . . . . . . . . . . . 14 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (^◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢 ∈ 𝐽) → Fun 𝐹)
37		toponss 13978	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ 𝑋)
38	19, 37	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦 ∈ 𝐾) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑥 ∈ (^◡𝐹 “ 𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ 𝑋)
39	35	fdmd 5391	. . . . . . . . . . . . . . 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)
40	38, 39	sseqtrrd 3209	. . . . . . . . . . . . . 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)
41		funimass3 5652	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ 𝑢 ⊆ dom 𝐹) → ((𝐹 “ 𝑢) ⊆ 𝑦 ↔ 𝑢 ⊆ (^◡𝐹 “ 𝑦)))
42	36, 40, 41	syl2anc 411	. . . . . . . . . . . . 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)))
43	42	anbi2d 464	. . . . . . . . . . . 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))))
44	43	rexbidva 2487	. . . . . . . . . . 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))))
45	34, 44	mpbid 147	. . . . . . . . . 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)))
46	45	expr 375	. . . . . . . . 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))))
47	46	ralimdva 2557	. . . . . . . 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))))
48	18, 47	syld 45	. . . . . . 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))))
49	48	impr 379	. . . . . 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)))
50	49	an32s 568	. . . . 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)))
51		topontop 13966	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
52	51	ad3antrrr 492	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦 ∈ 𝐾) → 𝐽 ∈ Top)
53		eltop2 14022	. . . . . 6 ⊢ (𝐽 ∈ Top → ((^◡𝐹 “ 𝑦) ∈ 𝐽 ↔ ∀𝑥 ∈ (^◡𝐹 “ 𝑦)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ 𝑦))))
54	52, 53	syl 14	. . . . 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))))
55	50, 54	mpbird 167	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦 ∈ 𝐾) → (^◡𝐹 “ 𝑦) ∈ 𝐽)
56	55	ralrimiva 2563	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∀𝑦 ∈ 𝐾 (^◡𝐹 “ 𝑦) ∈ 𝐽)
57	1	adantr 276	. . 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)))
58	12, 56, 57	mpbir2and 946	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾))
59	11, 58	impbida 596	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ∀wral 2468 ∃wrex 2469 ⊆ wss 3144 ∪ cuni 3824 ^◡ccnv 4643 dom cdm 4644 “ cima 4647 Fun wfun 5229 Fn wfn 5230 ⟶wf 5231 ‘cfv 5235 (class class class)co 5895 Topctop 13949 TopOnctopon 13962 Cn ccn 14137 CnP ccnp 14138

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-map 6675 df-topgen 12762 df-top 13950 df-topon 13963 df-cn 14140 df-cnp 14141

This theorem is referenced by: cncnp2m 14183 cnnei 14184 cnconst2 14185 metcn 14466 txmetcn 14471 cnlimcim 14592 cnlimc 14593 dvcn 14616

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