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

Theorem cnptoprest 12998
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 995 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐽 ∈ Top)
2 simpl3 997 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐴𝑋)
3 cnprest.1 . . . . . . . . . 10 𝑋 = 𝐽
43ntrss2 12880 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
51, 2, 4syl2anc 409 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
6 simprl 526 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝑃 ∈ ((int‘𝐽)‘𝐴))
75, 6sseldd 3148 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝑃𝐴)
8 fvres 5518 . . . . . . 7 (𝑃𝐴 → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
97, 8syl 14 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
109eqcomd 2176 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹𝑃) = ((𝐹𝐴)‘𝑃))
1110eleq1d 2239 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝐹𝑃) ∈ 𝑦 ↔ ((𝐹𝐴)‘𝑃) ∈ 𝑦))
12 inss1 3347 . . . . . . . . 9 (𝑥𝐴) ⊆ 𝑥
13 imass2 4985 . . . . . . . . 9 ((𝑥𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥))
14 sstr2 3154 . . . . . . . . 9 ((𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
1512, 13, 14mp2b 8 . . . . . . . 8 ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)
1615anim2i 340 . . . . . . 7 ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
1716reximi 2567 . . . . . 6 (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
183ntropn 12876 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
191, 2, 18syl2anc 409 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
20 inopn 12760 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑥𝐽 ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
21203com23 1204 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽𝑥𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
22213expia 1200 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
231, 19, 22syl2anc 409 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝑥𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
24 elin 3310 . . . . . . . . . . . . . 14 (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ↔ (𝑃𝑥𝑃 ∈ ((int‘𝐽)‘𝐴)))
2524simplbi2com 1437 . . . . . . . . . . . . 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 3353 . . . . . . . . . . . . . 14 (((int‘𝐽)‘𝐴) ⊆ 𝐴 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥𝐴))
28 imass2 4985 . . . . . . . . . . . . . 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 3154 . . . . . . . . . . . . 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 333 . . . . . . . . . . 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 333 . . . . . . . . . 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 2234 . . . . . . . . . . . 12 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝑃𝑧𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
35 imaeq2 4947 . . . . . . . . . . . . 13 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝐹𝑧) = (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
3635sseq1d 3176 . . . . . . . . . . . 12 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
3734, 36anbi12d 470 . . . . . . . . . . 11 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)))
3837rspcev 2834 . . . . . . . . . 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 2587 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
42 eleq2 2234 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑃𝑧𝑃𝑥))
43 imaeq2 4947 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
4443sseq1d 3176 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ 𝑦))
4542, 44anbi12d 470 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
4645cbvrexv 2697 . . . . . . 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 2733 . . . . . . . 8 𝑥 ∈ V
5049inex1 4121 . . . . . . 7 (𝑥𝐴) ∈ V
5150a1i 9 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
52 uniexg 4422 . . . . . . . . 9 (𝐽 ∈ Top → 𝐽 ∈ V)
531, 52syl 14 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐽 ∈ V)
542, 3sseqtrdi 3195 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐴 𝐽)
5553, 54ssexd 4127 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐴 ∈ V)
56 elrest 12579 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
571, 55, 56syl2anc 409 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
58 eleq2 2234 . . . . . . . 8 (𝑧 = (𝑥𝐴) → (𝑃𝑧𝑃 ∈ (𝑥𝐴)))
59 elin 3310 . . . . . . . . . 10 (𝑃 ∈ (𝑥𝐴) ↔ (𝑃𝑥𝑃𝐴))
6059rbaib 916 . . . . . . . . 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 460 . . . . . . 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 4951 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = ((𝐹𝐴) “ (𝑥𝐴)))
65 inss2 3348 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝐴
66 resima2 4923 . . . . . . . . . 10 ((𝑥𝐴) ⊆ 𝐴 → ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴)))
6765, 66ax-mp 5 . . . . . . . . 9 ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴))
6864, 67eqtrdi 2219 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = (𝐹 “ (𝑥𝐴)))
6968sseq1d 3176 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → (((𝐹𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
7062, 69anbi12d 470 . . . . . 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 4448 . . . . 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 2470 . 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 12775 . . . . 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 996 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐾 ∈ Top)
78 cnprest.2 . . . . . 6 𝑌 = 𝐾
7978toptopon 12775 . . . . 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 3148 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝑃𝑋)
82 iscnp 12958 . . . 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 1233 . . 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 527 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐹:𝑋𝑌)
8584biantrurd 303 . . 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 1016 . . . . . . 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 1017 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐴𝑋)
90 resttopon 12930 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
9188, 89, 90syl2anc 409 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
92 simp3 994 . . . . . 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 1013 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
95 simp2l 1018 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃 ∈ ((int‘𝐽)‘𝐴))
9694, 95sseldd 3148 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃𝐴)
97 iscnp 12958 . . . . 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 1233 . . . 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 1019 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐹:𝑋𝑌)
10099, 89fssresd 5372 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹𝐴):𝐴𝑌)
101100biantrurd 303 . . . 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 1244 . 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 973   = wceq 1348  wcel 2141  wral 2448  wrex 2449  Vcvv 2730  cin 3120  wss 3121   cuni 3794  cres 4611  cima 4612  wf 5192  cfv 5196  (class class class)co 5851  t crest 12572  Topctop 12754  TopOnctopon 12767  intcnt 12852   CnP ccnp 12945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-map 6626  df-rest 12574  df-topgen 12593  df-top 12755  df-topon 12768  df-bases 12800  df-ntr 12855  df-cnp 12948
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator