Theorem cnptoprest 13966

Description: Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 5-Apr-2023.)

Hypotheses

Ref	Expression
cnprest.1	⊢ 𝑋 = ∪ 𝐽
cnprest.2	⊢ 𝑌 = ∪ 𝐾

Assertion

Ref	Expression
cnptoprest	⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃)))

Proof of Theorem cnptoprest

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1001	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → 𝐽 ∈ Top)
2		simpl3 1003	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → 𝐴 ⊆ 𝑋)
3		cnprest.1	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
4	3	ntrss2 13848	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
5	1, 2, 4	syl2anc 411	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
6		simprl 529	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → 𝑃 ∈ ((int‘𝐽)‘𝐴))
7	5, 6	sseldd 3168	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → 𝑃 ∈ 𝐴)
8		fvres 5551	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑃) = (𝐹‘𝑃))
9	7, 8	syl 14	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → ((𝐹 ↾ 𝐴)‘𝑃) = (𝐹‘𝑃))
10	9	eqcomd 2193	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝐹‘𝑃) = ((𝐹 ↾ 𝐴)‘𝑃))
11	10	eleq1d 2256	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → ((𝐹‘𝑃) ∈ 𝑦 ↔ ((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦))
12		inss1 3367	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
13		imass2 5016	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ (𝐹 “ 𝑥))
14		sstr2 3174	. . . . . . . . 9 ⊢ ((𝐹 “ (𝑥 ∩ 𝐴)) ⊆ (𝐹 “ 𝑥) → ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦))
15	12, 13, 14	mp2b 8	. . . . . . . 8 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)
16	15	anim2i 342	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → (𝑃 ∈ 𝑥 ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦))
17	16	reximi 2584	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦))
18	3	ntropn 13844	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
19	1, 2, 18	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
20		inopn 13730	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
21	20	3com23 1210	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽 ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
22	21	3expia 1206	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥 ∈ 𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
23	1, 19, 22	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝑥 ∈ 𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
24		elin 3330	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ↔ (𝑃 ∈ 𝑥 ∧ 𝑃 ∈ ((int‘𝐽)‘𝐴)))
25	24	simplbi2com 1454	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ((int‘𝐽)‘𝐴) → (𝑃 ∈ 𝑥 → 𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
26	6, 25	syl 14	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝑃 ∈ 𝑥 → 𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
27		sslin 3373	. . . . . . . . . . . . . 14 ⊢ (((int‘𝐽)‘𝐴) ⊆ 𝐴 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥 ∩ 𝐴))
28		imass2 5016	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥 ∩ 𝐴) → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥 ∩ 𝐴)))
29	5, 27, 28	3syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥 ∩ 𝐴)))
30		sstr2 3174	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥 ∩ 𝐴)) → ((𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
31	29, 30	syl 14	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → ((𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
32	26, 31	anim12d 335	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦) → (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)))
33	23, 32	anim12d 335	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → ((𝑥 ∈ 𝐽 ∧ (𝑃 ∈ 𝑥 ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)) → ((𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))))
34		eleq2 2251	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
35		imaeq2 4978	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝐹 “ 𝑧) = (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
36	35	sseq1d 3196	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝐹 “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
37	34, 36	anbi12d 473	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)))
38	37	rspcev 2853	. . . . . . . . . 10 ⊢ (((𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)) → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦))
39	33, 38	syl6 33	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → ((𝑥 ∈ 𝐽 ∧ (𝑃 ∈ 𝑥 ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)) → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦)))
40	39	expdimp 259	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) ∧ 𝑥 ∈ 𝐽) → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦) → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦)))
41	40	rexlimdva 2604	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦) → ∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦)))
42		eleq2 2251	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ 𝑥))
43		imaeq2 4978	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝐹 “ 𝑧) = (𝐹 “ 𝑥))
44	43	sseq1d 3196	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝐹 “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ 𝑥) ⊆ 𝑦))
45	42, 44	anbi12d 473	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))
46	45	cbvrexv 2716	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐽 (𝑃 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))
47	41, 46	imbitrdi 161	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))
48	17, 47	impbid2 143	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)))
49		vex 2752	. . . . . . . 8 ⊢ 𝑥 ∈ V
50	49	inex1 4149	. . . . . . 7 ⊢ (𝑥 ∩ 𝐴) ∈ V
51	50	a1i 9	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ V)
52		uniexg 4451	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
53	1, 52	syl 14	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → ∪ 𝐽 ∈ V)
54	2, 3	sseqtrdi 3215	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → 𝐴 ⊆ ∪ 𝐽)
55	53, 54	ssexd 4155	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → 𝐴 ∈ V)
56		elrest 12712	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐴)))
57	1, 55, 56	syl2anc 411	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝑧 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐴)))
58		eleq2 2251	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∩ 𝐴) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ (𝑥 ∩ 𝐴)))
59		elin 3330	. . . . . . . . . 10 ⊢ (𝑃 ∈ (𝑥 ∩ 𝐴) ↔ (𝑃 ∈ 𝑥 ∧ 𝑃 ∈ 𝐴))
60	59	rbaib 922	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → (𝑃 ∈ (𝑥 ∩ 𝐴) ↔ 𝑃 ∈ 𝑥))
61	7, 60	syl 14	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝑃 ∈ (𝑥 ∩ 𝐴) ↔ 𝑃 ∈ 𝑥))
62	58, 61	sylan9bbr 463	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ 𝑥))
63		simpr 110	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → 𝑧 = (𝑥 ∩ 𝐴))
64	63	imaeq2d 4982	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → ((𝐹 ↾ 𝐴) “ 𝑧) = ((𝐹 ↾ 𝐴) “ (𝑥 ∩ 𝐴)))
65		inss2 3368	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
66		resima2 4953	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝐴) ⊆ 𝐴 → ((𝐹 ↾ 𝐴) “ (𝑥 ∩ 𝐴)) = (𝐹 “ (𝑥 ∩ 𝐴)))
67	65, 66	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) “ (𝑥 ∩ 𝐴)) = (𝐹 “ (𝑥 ∩ 𝐴))
68	64, 67	eqtrdi 2236	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → ((𝐹 ↾ 𝐴) “ 𝑧) = (𝐹 “ (𝑥 ∩ 𝐴)))
69	68	sseq1d 3196	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → (((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦))
70	62, 69	anbi12d 473	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → ((𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ 𝑥 ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)))
71	51, 57, 70	rexxfr2d 4477	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)))
72	48, 71	bitr4d 191	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) ↔ ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))
73	11, 72	imbi12d 234	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ↔ (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦))))
74	73	ralbidv 2487	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ↔ ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦))))
75	3	toptopon 13745	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
76	1, 75	sylib 122	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
77		simpl2 1002	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → 𝐾 ∈ Top)
78		cnprest.2	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
79	78	toptopon 13745	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
80	77, 79	sylib 122	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → 𝐾 ∈ (TopOn‘𝑌))
81	2, 7	sseldd 3168	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → 𝑃 ∈ 𝑋)
82		iscnp 13926	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
83	76, 80, 81, 82	syl3anc 1248	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
84		simprr 531	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → 𝐹:𝑋⟶𝑌)
85	84	biantrurd 305	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
86	83, 85	bitr4d 191	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))
87		simp1l 1022	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ Top)
88	87, 75	sylib 122	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ (TopOn‘𝑋))
89		simp1r 1023	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → 𝐴 ⊆ 𝑋)
90		resttopon 13898	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
91	88, 89, 90	syl2anc 411	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
92		simp3 1000	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
93	92, 79	sylib 122	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘𝑌))
94	4	3ad2ant1 1019	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
95		simp2l 1024	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → 𝑃 ∈ ((int‘𝐽)‘𝐴))
96	94, 95	sseldd 3168	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → 𝑃 ∈ 𝐴)
97		iscnp 13926	. . . . 5 ⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝐴) → ((𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹 ↾ 𝐴):𝐴⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))))
98	91, 93, 96, 97	syl3anc 1248	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → ((𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹 ↾ 𝐴):𝐴⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))))
99		simp2r 1025	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → 𝐹:𝑋⟶𝑌)
100	99, 89	fssresd 5404	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → (𝐹 ↾ 𝐴):𝐴⟶𝑌)
101	100	biantrurd 305	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → (∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)) ↔ ((𝐹 ↾ 𝐴):𝐴⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))))
102	98, 101	bitr4d 191	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐾 ∈ Top) → ((𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃) ↔ ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦))))
103	1, 2, 6, 84, 77, 102	syl221anc 1259	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → ((𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃) ↔ ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦))))
104	74, 86, 103	3bitr4d 220	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋⟶𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 979   = wceq 1363   ∈ wcel 2158  ∀wral 2465  ∃wrex 2466  Vcvv 2749   ∩ cin 3140   ⊆ wss 3141  ∪ cuni 3821   ↾ cres 4640   “ cima 4641  ⟶wf 5224  ‘cfv 5228  (class class class)co 5888   ↾_t crest 12705  Topctop 13724  TopOnctopon 13737  intcnt 13820   CnP ccnp 13913

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-map 6663  df-rest 12707  df-topgen 12726  df-top 13725  df-topon 13738  df-bases 13770  df-ntr 13823  df-cnp 13916

This theorem is referenced by: (None)