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Theorem cnpnei 15411

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 5150	. . . . . . . 8 ⊢ (^◡𝐹 “ 𝑦) ⊆ dom 𝐹
2		fdm 5539	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
3	1, 2	sseqtrid 3298	. . . . . . 7 ⊢ (𝐹:𝑋⟶𝑌 → (^◡𝐹 “ 𝑦) ⊆ 𝑋)
4	3	3ad2ant3 1051	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → (^◡𝐹 “ 𝑦) ⊆ 𝑋)
5	4	ad2antrr 492	. . . . 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 15341	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})) → ∃𝑔 ∈ 𝐾 ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))
7	6	3ad2antl2 1191	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})) → ∃𝑔 ∈ 𝐾 ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))
8	7	ad2ant2rl 515	. . . . . 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		cnpnei.1	. . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝐽
10	9	toptopon 15210	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
11	10	biimpi 120	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
12	11	3ad2ant1 1049	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
13	12	ad3antrrr 496	. . . . . . . 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))) → 𝐽 ∈ (TopOn‘𝑋))
14		cnpnei.2	. . . . . . . . . . . 12 ⊢ 𝑌 = ∪ 𝐾
15	14	toptopon 15210	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
16	15	biimpi 120	. . . . . . . . . 10 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘𝑌))
17	16	3ad2ant2 1050	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → 𝐾 ∈ (TopOn‘𝑌))
18	17	ad3antrrr 496	. . . . . . . 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))) → 𝐾 ∈ (TopOn‘𝑌))
19		simpllr 540	. . . . . . . 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))) → 𝐴 ∈ 𝑋)
20		simplrl 541	. . . . . . . 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))
21		simprl 535	. . . . . . . 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))) → 𝑔 ∈ 𝐾)
22		simprrl 545	. . . . . . . . 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))) → {(𝐹‘𝐴)} ⊆ 𝑔)
23		fvexg 5714	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ V)
24	20, 19, 23	syl2anc 415	. . . . . . . . . 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))) → (𝐹‘𝐴) ∈ V)
25		snssg 3849	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ V → ((𝐹‘𝐴) ∈ 𝑔 ↔ {(𝐹‘𝐴)} ⊆ 𝑔))
26	24, 25	syl 14	. . . . . . . . 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))) → ((𝐹‘𝐴) ∈ 𝑔 ↔ {(𝐹‘𝐴)} ⊆ 𝑔))
27	22, 26	mpbird 167	. . . . . . . 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‘𝐴) ∈ 𝑔)
28		icnpimaex 15403	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑔 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑔)) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ (𝐹 “ 𝑜) ⊆ 𝑔))
29	13, 18, 19, 20, 21, 27, 28	syl33anc 1293	. . . . . . 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))
30		sstr2 3255	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ 𝑜) ⊆ 𝑔 → (𝑔 ⊆ 𝑦 → (𝐹 “ 𝑜) ⊆ 𝑦))
31	30	com12 30	. . . . . . . . . . . 12 ⊢ (𝑔 ⊆ 𝑦 → ((𝐹 “ 𝑜) ⊆ 𝑔 → (𝐹 “ 𝑜) ⊆ 𝑦))
32	31	ad2antll 495	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)) → ((𝐹 “ 𝑜) ⊆ 𝑔 → (𝐹 “ 𝑜) ⊆ 𝑦))
33	32	ad2antlr 493	. . . . . . . . . 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))
34		ffun 5536	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
35	34	3ad2ant3 1051	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → Fun 𝐹)
36	35	ad2antrr 492	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → Fun 𝐹)
37	36	ad2antrr 492	. . . . . . . . . . 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)
38	9	eltopss 15201	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋)
39	38	adantlr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋)
40	2	sseq2d 3278	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑋⟶𝑌 → (𝑜 ⊆ dom 𝐹 ↔ 𝑜 ⊆ 𝑋))
41	40	ad2antlr 493	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → (𝑜 ⊆ dom 𝐹 ↔ 𝑜 ⊆ 𝑋))
42	39, 41	mpbird 167	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹)
43	42	3adantl2 1185	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹)
44	43	adantlr 481	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹)
45	44	adantlr 481	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹)
46	45	adantlr 481	. . . . . . . . . . 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)
47		funimass3 5825	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑜 ⊆ dom 𝐹) → ((𝐹 “ 𝑜) ⊆ 𝑦 ↔ 𝑜 ⊆ (^◡𝐹 “ 𝑦)))
48	37, 46, 47	syl2anc 415	. . . . . . . . . 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)))
49	33, 48	sylibd 149	. . . . . . . . 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)))
50	49	anim2d 337	. . . . . . . 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))))
51	50	reximdva 2652	. . . . . . 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))))
52	29, 51	mpd 13	. . . . . 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)))
53	8, 52	rexlimddv 2673	. . . . 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)))
54	9	isneip 15338	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) ↔ ((^◡𝐹 “ 𝑦) ⊆ 𝑋 ∧ ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (^◡𝐹 “ 𝑦)))))
55	54	3ad2antl1 1190	. . . . . 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)))))
56	55	adantr 276	. . . . 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)))))
57	5, 53, 56	mpbir2and 957	. . . 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}))
58	57	exp32 365	. . 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}))))
59	58	ralrimdv 2629	. 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})))
60		simpll3 1069	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) → 𝐹:𝑋⟶𝑌)
61		opnneip 15351	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Top ∧ 𝑜 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑜) → 𝑜 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))
62		imaeq2 5122	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑜 → (^◡𝐹 “ 𝑦) = (^◡𝐹 “ 𝑜))
63	62	eleq1d 2307	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑜 → ((^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) ↔ (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
64	63	rspcv 2925	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
65	61, 64	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝑜 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑜) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
66	65	3com23 1240	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Top ∧ (𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
67	66	3expb 1235	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(^◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})))
68	67	3ad2antl2 1191	. . . . . . . . . 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})))
69	68	adantlr 481	. . . . . . . . 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})))
70		neii2 15341	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})) → ∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (^◡𝐹 “ 𝑜)))
71	70	ex 115	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → ((^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (^◡𝐹 “ 𝑜))))
72	71	3ad2ant1 1049	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → ((^◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (^◡𝐹 “ 𝑜))))
73	72	ad2antrr 492	. . . . . . . . 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))))
74		snssg 3849	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ 𝑔 ↔ {𝐴} ⊆ 𝑔))
75	74	ad3antlr 497	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → (𝐴 ∈ 𝑔 ↔ {𝐴} ⊆ 𝑔))
76	35	ad3antrrr 496	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → Fun 𝐹)
77	9	eltopss 15201	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ 𝑋)
78	77	3ad2antl1 1190	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ 𝑋)
79	2	sseq2d 3278	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑋⟶𝑌 → (𝑔 ⊆ dom 𝐹 ↔ 𝑔 ⊆ 𝑋))
80	79	3ad2ant3 1051	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → (𝑔 ⊆ dom 𝐹 ↔ 𝑔 ⊆ 𝑋))
81	80	biimpar 297	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑔 ⊆ 𝑋) → 𝑔 ⊆ dom 𝐹)
82	78, 81	syldan 282	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ dom 𝐹)
83	82	adantlr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ dom 𝐹)
84	83	adantlr 481	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ dom 𝐹)
85		funimass3 5825	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑔 ⊆ dom 𝐹) → ((𝐹 “ 𝑔) ⊆ 𝑜 ↔ 𝑔 ⊆ (^◡𝐹 “ 𝑜)))
86	76, 84, 85	syl2anc 415	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → ((𝐹 “ 𝑔) ⊆ 𝑜 ↔ 𝑔 ⊆ (^◡𝐹 “ 𝑜)))
87	75, 86	anbi12d 477	. . . . . . . . . . 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))))
88	87	biimprd 158	. . . . . . . . . 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)))
89	88	reximdva 2652	. . . . . . . . 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)))
90	69, 73, 89	3syld 57	. . . . . . . 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)))
91	90	exp32 365	. . . . . . 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)))))
92	91	com24 87	. . . . . 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)))))
93	92	imp 124	. . . . 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))))
94	93	ralrimiv 2622	. . . 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)))
95	11	3ad2ant1 1049	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
96	16	3ad2ant2 1050	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
97		simp3 1030	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
98		iscnp 15391	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))))
99	95, 96, 97, 98	syl3anc 1278	. . . . . . 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)))))
100	99	3expa 1234	. . . . . 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)))))
101	100	3adantl3 1186	. . . . 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)))))
102	101	adantr 276	. . . 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)))))
103	60, 94, 102	mpbir2and 957	. . 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))
104	103	ex 115	. 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)))
105	59, 104	impbid 129	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 set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 Vcvv 2821 ⊆ wss 3220 {csn 3709 ∪ cuni 3935 ^◡ccnv 4773 dom cdm 4774 “ cima 4777 Fun wfun 5371 ⟶wf 5373 ‘cfv 5377 (class class class)co 6085 Topctop 15189 TopOnctopon 15202 neicnei 15330 CnP ccnp 15378

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-map 6924 df-top 15190 df-topon 15203 df-nei 15331 df-cnp 15381

This theorem is used by: (None)

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