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Theorem cnptoprest 13406
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 1000 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐽 ∈ Top)
2 simpl3 1002 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐴𝑋)
3 cnprest.1 . . . . . . . . . 10 𝑋 = 𝐽
43ntrss2 13288 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
51, 2, 4syl2anc 411 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
6 simprl 529 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝑃 ∈ ((int‘𝐽)‘𝐴))
75, 6sseldd 3156 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝑃𝐴)
8 fvres 5535 . . . . . . 7 (𝑃𝐴 → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
97, 8syl 14 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
109eqcomd 2183 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹𝑃) = ((𝐹𝐴)‘𝑃))
1110eleq1d 2246 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝐹𝑃) ∈ 𝑦 ↔ ((𝐹𝐴)‘𝑃) ∈ 𝑦))
12 inss1 3355 . . . . . . . . 9 (𝑥𝐴) ⊆ 𝑥
13 imass2 5000 . . . . . . . . 9 ((𝑥𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥))
14 sstr2 3162 . . . . . . . . 9 ((𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
1512, 13, 14mp2b 8 . . . . . . . 8 ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)
1615anim2i 342 . . . . . . 7 ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
1716reximi 2574 . . . . . 6 (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
183ntropn 13284 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
191, 2, 18syl2anc 411 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
20 inopn 13168 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑥𝐽 ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
21203com23 1209 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽𝑥𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
22213expia 1205 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
231, 19, 22syl2anc 411 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝑥𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
24 elin 3318 . . . . . . . . . . . . . 14 (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ↔ (𝑃𝑥𝑃 ∈ ((int‘𝐽)‘𝐴)))
2524simplbi2com 1444 . . . . . . . . . . . . 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 3361 . . . . . . . . . . . . . 14 (((int‘𝐽)‘𝐴) ⊆ 𝐴 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥𝐴))
28 imass2 5000 . . . . . . . . . . . . . 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 3162 . . . . . . . . . . . . 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 335 . . . . . . . . . . 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 335 . . . . . . . . . 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 2241 . . . . . . . . . . . 12 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝑃𝑧𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
35 imaeq2 4962 . . . . . . . . . . . . 13 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝐹𝑧) = (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
3635sseq1d 3184 . . . . . . . . . . . 12 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
3734, 36anbi12d 473 . . . . . . . . . . 11 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)))
3837rspcev 2841 . . . . . . . . . 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 259 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑥𝐽) → ((𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
4140rexlimdva 2594 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
42 eleq2 2241 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑃𝑧𝑃𝑥))
43 imaeq2 4962 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
4443sseq1d 3184 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ 𝑦))
4542, 44anbi12d 473 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
4645cbvrexv 2704 . . . . . . 7 (∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
4741, 46syl6ib 161 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
4817, 47impbid2 143 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
49 vex 2740 . . . . . . . 8 𝑥 ∈ V
5049inex1 4134 . . . . . . 7 (𝑥𝐴) ∈ V
5150a1i 9 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
52 uniexg 4436 . . . . . . . . 9 (𝐽 ∈ Top → 𝐽 ∈ V)
531, 52syl 14 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐽 ∈ V)
542, 3sseqtrdi 3203 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐴 𝐽)
5553, 54ssexd 4140 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐴 ∈ V)
56 elrest 12643 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
571, 55, 56syl2anc 411 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
58 eleq2 2241 . . . . . . . 8 (𝑧 = (𝑥𝐴) → (𝑃𝑧𝑃 ∈ (𝑥𝐴)))
59 elin 3318 . . . . . . . . . 10 (𝑃 ∈ (𝑥𝐴) ↔ (𝑃𝑥𝑃𝐴))
6059rbaib 921 . . . . . . . . 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 463 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → (𝑃𝑧𝑃𝑥))
63 simpr 110 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → 𝑧 = (𝑥𝐴))
6463imaeq2d 4966 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = ((𝐹𝐴) “ (𝑥𝐴)))
65 inss2 3356 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝐴
66 resima2 4937 . . . . . . . . . 10 ((𝑥𝐴) ⊆ 𝐴 → ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴)))
6765, 66ax-mp 5 . . . . . . . . 9 ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴))
6864, 67eqtrdi 2226 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = (𝐹 “ (𝑥𝐴)))
6968sseq1d 3184 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) ∧ 𝑧 = (𝑥𝐴)) → (((𝐹𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
7062, 69anbi12d 473 . . . . . 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 4462 . . . . 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 191 . . . 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 234 . . 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 2477 . 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 13183 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
761, 75sylib 122 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
77 simpl2 1001 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐾 ∈ Top)
78 cnprest.2 . . . . . 6 𝑌 = 𝐾
7978toptopon 13183 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
8077, 79sylib 122 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐾 ∈ (TopOn‘𝑌))
812, 7sseldd 3156 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝑃𝑋)
82 iscnp 13366 . . . 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 1238 . . 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 531 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → 𝐹:𝑋𝑌)
8584biantrurd 305 . . 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 191 . 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 1021 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ Top)
8887, 75sylib 122 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ (TopOn‘𝑋))
89 simp1r 1022 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐴𝑋)
90 resttopon 13338 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
9188, 89, 90syl2anc 411 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
92 simp3 999 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
9392, 79sylib 122 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘𝑌))
9443ad2ant1 1018 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
95 simp2l 1023 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃 ∈ ((int‘𝐽)‘𝐴))
9694, 95sseldd 3156 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃𝐴)
97 iscnp 13366 . . . . 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 1238 . . . 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 1024 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐹:𝑋𝑌)
10099, 89fssresd 5388 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹𝐴):𝐴𝑌)
101100biantrurd 305 . . . 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 191 . . 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 1249 . 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 220 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 104  wb 105  w3a 978   = wceq 1353  wcel 2148  wral 2455  wrex 2456  Vcvv 2737  cin 3128  wss 3129   cuni 3807  cres 4625  cima 4626  wf 5208  cfv 5212  (class class class)co 5869  t crest 12636  Topctop 13162  TopOnctopon 13175  intcnt 13260   CnP ccnp 13353
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-rest 12638  df-topgen 12657  df-top 13163  df-topon 13176  df-bases 13208  df-ntr 13263  df-cnp 13356
This theorem is referenced by: (None)
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