Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnptoprest GIF version

Theorem cnptoprest 12189
 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 ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃)))

Proof of Theorem cnptoprest
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 952 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐽 ∈ Top)
2 simpl3 954 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐴𝑋)
3 cnprest.1 . . . . . . . . . 10 𝑋 = 𝐽
43ntrss2 12072 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
51, 2, 4syl2anc 406 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
6 simprl 501 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝑃 ∈ ((int‘𝐽)‘𝐴))
75, 6sseldd 3048 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝑃𝐴)
8 fvres 5377 . . . . . . 7 (𝑃𝐴 → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
97, 8syl 14 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
109eqcomd 2105 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹𝑃) = ((𝐹𝐴)‘𝑃))
1110eleq1d 2168 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝐹𝑃) ∈ 𝑦 ↔ ((𝐹𝐴)‘𝑃) ∈ 𝑦))
12 inss1 3243 . . . . . . . . 9 (𝑥𝐴) ⊆ 𝑥
13 imass2 4851 . . . . . . . . 9 ((𝑥𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥))
14 sstr2 3054 . . . . . . . . 9 ((𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
1512, 13, 14mp2b 8 . . . . . . . 8 ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)
1615anim2i 337 . . . . . . 7 ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
1716reximi 2488 . . . . . 6 (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
183ntropn 12068 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
191, 2, 18syl2anc 406 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
20 inopn 11952 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑥𝐽 ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
21203com23 1155 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽𝑥𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
22213expia 1151 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
231, 19, 22syl2anc 406 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝑥𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
24 elin 3206 . . . . . . . . . . . . . 14 (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ↔ (𝑃𝑥𝑃 ∈ ((int‘𝐽)‘𝐴)))
2524simplbi2com 1388 . . . . . . . . . . . . 13 (𝑃 ∈ ((int‘𝐽)‘𝐴) → (𝑃𝑥𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
266, 25syl 14 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝑃𝑥𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
27 sslin 3249 . . . . . . . . . . . . . 14 (((int‘𝐽)‘𝐴) ⊆ 𝐴 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥𝐴))
28 imass2 4851 . . . . . . . . . . . . . 14 ((𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥𝐴) → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥𝐴)))
295, 27, 283syl 17 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥𝐴)))
30 sstr2 3054 . . . . . . . . . . . . 13 ((𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥𝐴)) → ((𝐹 “ (𝑥𝐴)) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
3129, 30syl 14 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝐹 “ (𝑥𝐴)) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
3226, 31anim12d 331 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)))
3323, 32anim12d 331 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)) → ((𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))))
34 eleq2 2163 . . . . . . . . . . . 12 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝑃𝑧𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
35 imaeq2 4813 . . . . . . . . . . . . 13 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝐹𝑧) = (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
3635sseq1d 3076 . . . . . . . . . . . 12 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
3734, 36anbi12d 460 . . . . . . . . . . 11 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)))
3837rspcev 2744 . . . . . . . . . 10 (((𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦))
3933, 38syl6 33 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
4039expdimp 257 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑥𝐽) → ((𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
4140rexlimdva 2508 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
42 eleq2 2163 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑃𝑧𝑃𝑥))
43 imaeq2 4813 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
4443sseq1d 3076 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ 𝑦))
4542, 44anbi12d 460 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
4645cbvrexv 2613 . . . . . . 7 (∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
4741, 46syl6ib 160 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
4817, 47impbid2 142 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
49 vex 2644 . . . . . . . 8 𝑥 ∈ V
5049inex1 4002 . . . . . . 7 (𝑥𝐴) ∈ V
5150a1i 9 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
52 uniexg 4299 . . . . . . . . 9 (𝐽 ∈ Top → 𝐽 ∈ V)
531, 52syl 14 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐽 ∈ V)
542, 3syl6sseq 3095 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐴 𝐽)
5553, 54ssexd 4008 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐴 ∈ V)
56 elrest 11909 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
571, 55, 56syl2anc 406 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
58 eleq2 2163 . . . . . . . 8 (𝑧 = (𝑥𝐴) → (𝑃𝑧𝑃 ∈ (𝑥𝐴)))
59 elin 3206 . . . . . . . . . 10 (𝑃 ∈ (𝑥𝐴) ↔ (𝑃𝑥𝑃𝐴))
6059rbaib 874 . . . . . . . . 9 (𝑃𝐴 → (𝑃 ∈ (𝑥𝐴) ↔ 𝑃𝑥))
617, 60syl 14 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝑃 ∈ (𝑥𝐴) ↔ 𝑃𝑥))
6258, 61sylan9bbr 454 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → (𝑃𝑧𝑃𝑥))
63 simpr 109 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → 𝑧 = (𝑥𝐴))
6463imaeq2d 4817 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = ((𝐹𝐴) “ (𝑥𝐴)))
65 inss2 3244 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝐴
66 resima2 4789 . . . . . . . . . 10 ((𝑥𝐴) ⊆ 𝐴 → ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴)))
6765, 66ax-mp 7 . . . . . . . . 9 ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴))
6864, 67syl6eq 2148 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = (𝐹 “ (𝑥𝐴)))
6968sseq1d 3076 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → (((𝐹𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
7062, 69anbi12d 460 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → ((𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
7151, 57, 70rexxfr2d 4324 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
7248, 71bitr4d 190 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
7311, 72imbi12d 233 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦))))
7473ralbidv 2396 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦))))
753toptopon 11967 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
761, 75sylib 121 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
77 simpl2 953 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐾 ∈ Top)
78 cnprest.2 . . . . . 6 𝑌 = 𝐾
7978toptopon 11967 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
8077, 79sylib 121 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐾 ∈ (TopOn‘𝑌))
812, 7sseldd 3048 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝑃𝑋)
82 iscnp 12149 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
8376, 80, 81, 82syl3anc 1184 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
84 simprr 502 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐹:𝑋𝑌)
8584biantrurd 301 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
8683, 85bitr4d 190 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))))
87 simp1l 973 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ Top)
8887, 75sylib 121 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ (TopOn‘𝑋))
89 simp1r 974 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐴𝑋)
90 resttopon 12122 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
9188, 89, 90syl2anc 406 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
92 simp3 951 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
9392, 79sylib 121 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘𝑌))
9443ad2ant1 970 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
95 simp2l 975 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃 ∈ ((int‘𝐽)‘𝐴))
9694, 95sseldd 3048 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃𝐴)
97 iscnp 12149 . . . . 5 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝐴) → ((𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹𝐴):𝐴𝑌 ∧ ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))))
9891, 93, 96, 97syl3anc 1184 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹𝐴):𝐴𝑌 ∧ ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))))
99 simp2r 976 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐹:𝑋𝑌)
10099, 89fssresd 5235 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹𝐴):𝐴𝑌)
101100biantrurd 301 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)) ↔ ((𝐹𝐴):𝐴𝑌 ∧ ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))))
10298, 101bitr4d 190 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃) ↔ ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦))))
1031, 2, 6, 84, 77, 102syl221anc 1195 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃) ↔ ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦))))
10474, 86, 1033bitr4d 219 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 930   = wceq 1299   ∈ wcel 1448  ∀wral 2375  ∃wrex 2376  Vcvv 2641   ∩ cin 3020   ⊆ wss 3021  ∪ cuni 3683   ↾ cres 4479   “ cima 4480  ⟶wf 5055  ‘cfv 5059  (class class class)co 5706   ↾t crest 11902  Topctop 11946  TopOnctopon 11959  intcnt 12044   CnP ccnp 12137 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390 This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-map 6474  df-rest 11904  df-topgen 11923  df-top 11947  df-topon 11960  df-bases 11992  df-ntr 12047  df-cnp 12140 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator