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Theorem cnprest 21141
Description: Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.)
Hypotheses
Ref Expression
cnprest.1 𝑋 = 𝐽
cnprest.2 𝑌 = 𝐾
Assertion
Ref Expression
cnprest (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃)))

Proof of Theorem cnprest
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnptop2 21095 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
21a1i 11 . 2 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top))
3 cnptop2 21095 . . 3 ((𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
43a1i 11 . 2 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → ((𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃) → 𝐾 ∈ Top))
5 cnprest.1 . . . . . . . . . . . 12 𝑋 = 𝐽
65ntrss2 20909 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
763ad2ant1 1102 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
8 simp2l 1107 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃 ∈ ((int‘𝐽)‘𝐴))
97, 8sseldd 3637 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃𝐴)
10 fvres 6245 . . . . . . . . 9 (𝑃𝐴 → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
119, 10syl 17 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
1211eqcomd 2657 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹𝑃) = ((𝐹𝐴)‘𝑃))
1312eleq1d 2715 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝐹𝑃) ∈ 𝑦 ↔ ((𝐹𝐴)‘𝑃) ∈ 𝑦))
14 inss1 3866 . . . . . . . . . . 11 (𝑥𝐴) ⊆ 𝑥
15 imass2 5536 . . . . . . . . . . 11 ((𝑥𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥))
16 sstr2 3643 . . . . . . . . . . 11 ((𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
1714, 15, 16mp2b 10 . . . . . . . . . 10 ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)
1817anim2i 592 . . . . . . . . 9 ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
1918reximi 3040 . . . . . . . 8 (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
20 simp1l 1105 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ Top)
215ntropn 20901 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
22213ad2ant1 1102 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
23 inopn 20752 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝑥𝐽 ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
24233com23 1291 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽𝑥𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
25243expia 1286 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
2620, 22, 25syl2anc 694 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝑥𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
27 elin 3829 . . . . . . . . . . . . . . . 16 (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ↔ (𝑃𝑥𝑃 ∈ ((int‘𝐽)‘𝐴)))
2827simplbi2com 656 . . . . . . . . . . . . . . 15 (𝑃 ∈ ((int‘𝐽)‘𝐴) → (𝑃𝑥𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
298, 28syl 17 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝑃𝑥𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
30 sslin 3872 . . . . . . . . . . . . . . . . 17 (((int‘𝐽)‘𝐴) ⊆ 𝐴 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥𝐴))
317, 30syl 17 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥𝐴))
32 imass2 5536 . . . . . . . . . . . . . . . 16 ((𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥𝐴) → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥𝐴)))
3331, 32syl 17 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥𝐴)))
34 sstr2 3643 . . . . . . . . . . . . . . 15 ((𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥𝐴)) → ((𝐹 “ (𝑥𝐴)) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
3533, 34syl 17 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝐹 “ (𝑥𝐴)) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
3629, 35anim12d 585 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)))
3726, 36anim12d 585 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)) → ((𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))))
38 eleq2 2719 . . . . . . . . . . . . . 14 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝑃𝑧𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
39 imaeq2 5497 . . . . . . . . . . . . . . 15 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝐹𝑧) = (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
4039sseq1d 3665 . . . . . . . . . . . . . 14 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
4138, 40anbi12d 747 . . . . . . . . . . . . 13 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)))
4241rspcev 3340 . . . . . . . . . . . 12 (((𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦))
4337, 42syl6 35 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
4443expdimp 452 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑥𝐽) → ((𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
4544rexlimdva 3060 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
46 eleq2 2719 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑃𝑧𝑃𝑥))
47 imaeq2 5497 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
4847sseq1d 3665 . . . . . . . . . . 11 (𝑧 = 𝑥 → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ 𝑦))
4946, 48anbi12d 747 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
5049cbvrexv 3202 . . . . . . . . 9 (∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
5145, 50syl6ib 241 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
5219, 51impbid2 216 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
53 vex 3234 . . . . . . . . . 10 𝑥 ∈ V
5453inex1 4832 . . . . . . . . 9 (𝑥𝐴) ∈ V
5554a1i 11 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
56 uniexg 6997 . . . . . . . . . . 11 (𝐽 ∈ Top → 𝐽 ∈ V)
5720, 56syl 17 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ V)
58 simp1r 1106 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐴𝑋)
5958, 5syl6sseq 3684 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐴 𝐽)
6057, 59ssexd 4838 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐴 ∈ V)
61 elrest 16135 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
6220, 60, 61syl2anc 694 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
63 eleq2 2719 . . . . . . . . . 10 (𝑧 = (𝑥𝐴) → (𝑃𝑧𝑃 ∈ (𝑥𝐴)))
64 elin 3829 . . . . . . . . . . . 12 (𝑃 ∈ (𝑥𝐴) ↔ (𝑃𝑥𝑃𝐴))
6564rbaib 967 . . . . . . . . . . 11 (𝑃𝐴 → (𝑃 ∈ (𝑥𝐴) ↔ 𝑃𝑥))
669, 65syl 17 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝑃 ∈ (𝑥𝐴) ↔ 𝑃𝑥))
6763, 66sylan9bbr 737 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → (𝑃𝑧𝑃𝑥))
68 simpr 476 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → 𝑧 = (𝑥𝐴))
6968imaeq2d 5501 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = ((𝐹𝐴) “ (𝑥𝐴)))
70 inss2 3867 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝐴
71 resima2 5467 . . . . . . . . . . . 12 ((𝑥𝐴) ⊆ 𝐴 → ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴)))
7270, 71ax-mp 5 . . . . . . . . . . 11 ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴))
7369, 72syl6eq 2701 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = (𝐹 “ (𝑥𝐴)))
7473sseq1d 3665 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → (((𝐹𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
7567, 74anbi12d 747 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → ((𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
7655, 62, 75rexxfr2d 4913 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
7752, 76bitr4d 271 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
7813, 77imbi12d 333 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦))))
7978ralbidv 3015 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦))))
80 simp3 1083 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
8158, 9sseldd 3637 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃𝑋)
82 cnprest.2 . . . . . . . 8 𝑌 = 𝐾
835, 82iscnp2 21091 . . . . . . 7 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
8483baib 964 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
8520, 80, 81, 84syl3anc 1366 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
86 simp2r 1108 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐹:𝑋𝑌)
8786biantrurd 528 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
8885, 87bitr4d 271 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))))
895toptopon 20770 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
9020, 89sylib 208 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ (TopOn‘𝑋))
91 resttopon 21013 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
9290, 58, 91syl2anc 694 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
9382toptopon 20770 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
9480, 93sylib 208 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘𝑌))
95 iscnp 21089 . . . . . 6 (((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)))))
9692, 94, 9, 95syl3anc 1366 . . . . 5 (((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)))))
9786, 58fssresd 6109 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹𝐴):𝐴𝑌)
9897biantrurd 528 . . . . 5 (((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)))))
9996, 98bitr4d 271 . . . 4 (((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))))
10079, 88, 993bitr4d 300 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃)))
1011003expia 1286 . 2 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐾 ∈ Top → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃))))
1022, 4, 101pm5.21ndd 368 1 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607   cuni 4468  cres 5145  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  t crest 16128  Topctop 20746  TopOnctopon 20763  intcnt 20869   CnP ccnp 21077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-ntr 20872  df-cnp 21080
This theorem is referenced by:  limcres  23695  dvcnvrelem2  23826  psercn  24225  abelth  24240  cxpcn3  24534  efrlim  24741  cvmlift2lem11  31421  cvmlift2lem12  31422  cvmlift3lem7  31433  cncfuni  40417  cncfiooicclem1  40424  dirkercncflem4  40641  fourierdlem62  40703
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