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Theorem cnpnei 22161

Description: A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)

**Hypotheses**
Ref	Expression
cnpnei.1	⊢ 𝑋 = ∪ 𝐽
cnpnei.2	⊢ 𝑌 = ∪ 𝐾

**Assertion**
Ref	Expression
cnpnei	⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})))

Distinct variable groups: 𝑦,𝐴 𝑦,𝐹 𝑦,𝐽 𝑦,𝐾 𝑦,𝑋 𝑦,𝑌

Proof of Theorem cnpnei Dummy variables 𝑔 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvimass 5949	. . . . . . . 8 ⊢ (^◡𝐹 “ 𝑦) ⊆ dom 𝐹
2		fdm 6554	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
3	1, 2	sseqtrid 3953	. . . . . . 7 ⊢ (𝐹:𝑋⟶𝑌 → (^◡𝐹 “ 𝑦) ⊆ 𝑋)
4	3	3ad2ant3 1137	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → (^◡𝐹 “ 𝑦) ⊆ 𝑋)
5	4	ad2antrr 726	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → (^◡𝐹 “ 𝑦) ⊆ 𝑋)
6		neii2 22005	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})) → ∃𝑔 ∈ 𝐾 ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))
7	6	3ad2antl2 1188	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})) → ∃𝑔 ∈ 𝐾 ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))
8	7	ad2ant2rl 749	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → ∃𝑔 ∈ 𝐾 ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))
9		simpll 767	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
10		simprl 771	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → 𝑔 ∈ 𝐾)
11		fvex 6730	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝐴) ∈ V
12	11	snss 4699	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝐴) ∈ 𝑔 ↔ {(𝐹‘𝐴)} ⊆ 𝑔)
13	12	biimpri 231	. . . . . . . . . . . 12 ⊢ ({(𝐹‘𝐴)} ⊆ 𝑔 → (𝐹‘𝐴) ∈ 𝑔)
14	13	adantr 484	. . . . . . . . . . 11 ⊢ (({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦) → (𝐹‘𝐴) ∈ 𝑔)
15	14	ad2antll 729	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → (𝐹‘𝐴) ∈ 𝑔)
16	9, 10, 15	3jca 1130	. . . . . . . . 9 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑔 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑔))
17	16	adantll 714	. . . . . . . 8 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑔 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑔))
18		cnpimaex 22153	. . . . . . . 8 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑔 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑔) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ (𝐹 “ 𝑜) ⊆ 𝑔))
19	17, 18	syl 17	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ (𝐹 “ 𝑜) ⊆ 𝑔))
20		sstr2 3908	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ 𝑜) ⊆ 𝑔 → (𝑔 ⊆ 𝑦 → (𝐹 “ 𝑜) ⊆ 𝑦))
21	20	com12 32	. . . . . . . . . . . 12 ⊢ (𝑔 ⊆ 𝑦 → ((𝐹 “ 𝑜) ⊆ 𝑔 → (𝐹 “ 𝑜) ⊆ 𝑦))
22	21	ad2antll 729	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)) → ((𝐹 “ 𝑜) ⊆ 𝑔 → (𝐹 “ 𝑜) ⊆ 𝑦))
23	22	ad2antlr 727	. . . . . . . . . 10 ⊢ ((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → ((𝐹 “ 𝑜) ⊆ 𝑔 → (𝐹 “ 𝑜) ⊆ 𝑦))
24		ffun 6548	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
25	24	3ad2ant3 1137	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → Fun 𝐹)
26	25	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → Fun 𝐹)
27	26	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → Fun 𝐹)
28		cnpnei.1	. . . . . . . . . . . . . . . . 17 ⊢ 𝑋 = ∪ 𝐽
29	28	eltopss 21804	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋)
30	29	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋)
31	2	sseq2d 3933	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋⟶𝑌 → (𝑜 ⊆ dom 𝐹 ↔ 𝑜 ⊆ 𝑋))
32	31	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → (𝑜 ⊆ dom 𝐹 ↔ 𝑜 ⊆ 𝑋))
33	30, 32	mpbird 260	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹)
34	33	3adantl2 1169	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹)
35	34	adantlr 715	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹)
36	35	ad4ant14 752	. . . . . . . . . . 11 ⊢ ((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹)
37		funimass3 6874	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑜 ⊆ dom 𝐹) → ((𝐹 “ 𝑜) ⊆ 𝑦 ↔ 𝑜 ⊆ (^◡𝐹 “ 𝑦)))
38	27, 36, 37	syl2anc 587	. . . . . . . . . 10 ⊢ ((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → ((𝐹 “ 𝑜) ⊆ 𝑦 ↔ 𝑜 ⊆ (^◡𝐹 “ 𝑦)))
39	23, 38	sylibd 242	. . . . . . . . 9 ⊢ ((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → ((𝐹 “ 𝑜) ⊆ 𝑔 → 𝑜 ⊆ (^◡𝐹 “ 𝑦)))
40	39	anim2d 615	. . . . . . . 8 ⊢ ((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → ((𝐴 ∈ 𝑜 ∧ (𝐹 “ 𝑜) ⊆ 𝑔) → (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (^◡𝐹 “ 𝑦))))
41	40	reximdva 3193	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → (∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ (𝐹 “ 𝑜) ⊆ 𝑔) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (^◡𝐹 “ 𝑦))))
42	19, 41	mpd 15	. . . . . 6 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (^◡𝐹 “ 𝑦)))
43	8, 42	rexlimddv 3210	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (^◡𝐹 “ 𝑦)))
44	28	isneip 22002	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) ↔ ((^◡𝐹 “ 𝑦) ⊆ 𝑋 ∧ ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (^◡𝐹 “ 𝑦)))))
45	44	3ad2antl1 1187	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → ((^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) ↔ ((^◡𝐹 “ 𝑦) ⊆ 𝑋 ∧ ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (^◡𝐹 “ 𝑦)))))
46	45	adantr 484	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → ((^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) ↔ ((^◡𝐹 “ 𝑦) ⊆ 𝑋 ∧ ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (^◡𝐹 “ 𝑦)))))
47	5, 43, 46	mpbir2and 713	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → (^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}))
48	47	exp32 424	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}) → (^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}))))
49	48	ralrimdv 3109	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})))
50		simpll3 1216	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) → 𝐹:𝑋⟶𝑌)
51		opnneip 22016	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Top ∧ 𝑜 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑜) → 𝑜 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))
52		imaeq2 5925	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑜 → (^◡𝐹 “ 𝑦) = (^◡𝐹 “ 𝑜))
53	52	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑜 → ((^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) ↔ (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
54	53	rspcv 3532	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
55	51, 54	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑜 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑜) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
56	55	3com23 1128	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Top ∧ (𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
57	56	3expb 1122	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
58	57	3ad2antl2 1188	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
59	58	adantlr 715	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
60		neii2 22005	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})) → ∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (^◡𝐹 “ 𝑜)))
61	60	ex 416	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → ((^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (^◡𝐹 “ 𝑜))))
62	61	3ad2ant1 1135	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → ((^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (^◡𝐹 “ 𝑜))))
63	62	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → ((^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (^◡𝐹 “ 𝑜))))
64		snssg 4698	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ 𝑔 ↔ {𝐴} ⊆ 𝑔))
65	64	ad3antlr 731	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → (𝐴 ∈ 𝑔 ↔ {𝐴} ⊆ 𝑔))
66	25	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → Fun 𝐹)
67	28	eltopss 21804	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ 𝑋)
68	67	3ad2antl1 1187	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ 𝑋)
69	2	sseq2d 3933	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑋⟶𝑌 → (𝑔 ⊆ dom 𝐹 ↔ 𝑔 ⊆ 𝑋))
70	69	3ad2ant3 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → (𝑔 ⊆ dom 𝐹 ↔ 𝑔 ⊆ 𝑋))
71	70	biimpar 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑔 ⊆ 𝑋) → 𝑔 ⊆ dom 𝐹)
72	68, 71	syldan 594	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ dom 𝐹)
73	72	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ dom 𝐹)
74		funimass3 6874	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑔 ⊆ dom 𝐹) → ((𝐹 “ 𝑔) ⊆ 𝑜 ↔ 𝑔 ⊆ (^◡𝐹 “ 𝑜)))
75	66, 73, 74	syl2anc 587	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → ((𝐹 “ 𝑔) ⊆ 𝑜 ↔ 𝑔 ⊆ (^◡𝐹 “ 𝑜)))
76	65, 75	anbi12d 634	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → ((𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜) ↔ ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (^◡𝐹 “ 𝑜))))
77	76	biimprd 251	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → (({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (^◡𝐹 “ 𝑜)) → (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))
78	77	reximdva 3193	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → (∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (^◡𝐹 “ 𝑜)) → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))
79	59, 63, 78	3syld 60	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))
80	79	exp32 424	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴) ∈ 𝑜 → (𝑜 ∈ 𝐾 → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))))
81	80	com24 95	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (𝑜 ∈ 𝐾 → ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))))
82	81	imp 410	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) → (𝑜 ∈ 𝐾 → ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))))
83	82	ralrimiv 3104	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) → ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))
84		cnpnei.2	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐾
85	28, 84	iscnp2 22136	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ∈ 𝑋) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))))
86	85	baib 539	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))))
87	86	3expa 1120	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))))
88	87	3adantl3 1170	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))))
89	88	adantr 484	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))))
90	50, 83, 89	mpbir2and 713	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
91	90	ex 416	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)))
92	49, 91	impbid 215	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 ⊆ wss 3866 {csn 4541 ∪ cuni 4819 ^◡ccnv 5550 dom cdm 5551 “ cima 5554 Fun wfun 6374 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 Topctop 21790 neicnei 21994 CnP ccnp 22122

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-map 8510 df-top 21791 df-topon 21808 df-nei 21995 df-cnp 22125

This theorem is referenced by: (None)

W3C validator