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Theorem cnprest 22640
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 22594 . . 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 22594 . . 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 22408 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
763ad2ant1 1133 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
8 simp2l 1199 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃 ∈ ((int‘𝐽)‘𝐴))
97, 8sseldd 3945 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃𝐴)
109fvresd 6862 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
1110eqcomd 2742 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹𝑃) = ((𝐹𝐴)‘𝑃))
1211eleq1d 2822 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝐹𝑃) ∈ 𝑦 ↔ ((𝐹𝐴)‘𝑃) ∈ 𝑦))
13 inss1 4188 . . . . . . . . . . 11 (𝑥𝐴) ⊆ 𝑥
14 imass2 6054 . . . . . . . . . . 11 ((𝑥𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥))
15 sstr2 3951 . . . . . . . . . . 11 ((𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥) → ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
1613, 14, 15mp2b 10 . . . . . . . . . 10 ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)
1716anim2i 617 . . . . . . . . 9 ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
1817reximi 3087 . . . . . . . 8 (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
19 simp1l 1197 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ Top)
205ntropn 22400 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
21203ad2ant1 1133 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
22 inopn 22248 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝑥𝐽 ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
23223com23 1126 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽𝑥𝐽) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽)
24233expia 1121 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽) → (𝑥𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
2519, 21, 24syl2anc 584 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝑥𝐽 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽))
26 elin 3926 . . . . . . . . . . . . . . . 16 (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ↔ (𝑃𝑥𝑃 ∈ ((int‘𝐽)‘𝐴)))
2726simplbi2com 503 . . . . . . . . . . . . . . 15 (𝑃 ∈ ((int‘𝐽)‘𝐴) → (𝑃𝑥𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
288, 27syl 17 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝑃𝑥𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
29 sslin 4194 . . . . . . . . . . . . . . . . 17 (((int‘𝐽)‘𝐴) ⊆ 𝐴 → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥𝐴))
307, 29syl 17 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥𝐴))
31 imass2 6054 . . . . . . . . . . . . . . . 16 ((𝑥 ∩ ((int‘𝐽)‘𝐴)) ⊆ (𝑥𝐴) → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥𝐴)))
3230, 31syl 17 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥𝐴)))
33 sstr2 3951 . . . . . . . . . . . . . . 15 ((𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ (𝐹 “ (𝑥𝐴)) → ((𝐹 “ (𝑥𝐴)) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
3432, 33syl 17 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝐹 “ (𝑥𝐴)) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
3528, 34anim12d 609 . . . . . . . . . . . . 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)))
3625, 35anim12d 609 . . . . . . . . . . . 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))))
37 eleq2 2826 . . . . . . . . . . . . . 14 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝑃𝑧𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
38 imaeq2 6009 . . . . . . . . . . . . . . 15 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → (𝐹𝑧) = (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))))
3938sseq1d 3975 . . . . . . . . . . . . . 14 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦))
4037, 39anbi12d 631 . . . . . . . . . . . . 13 (𝑧 = (𝑥 ∩ ((int‘𝐽)‘𝐴)) → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)))
4140rspcev 3581 . . . . . . . . . . . 12 (((𝑥 ∩ ((int‘𝐽)‘𝐴)) ∈ 𝐽 ∧ (𝑃 ∈ (𝑥 ∩ ((int‘𝐽)‘𝐴)) ∧ (𝐹 “ (𝑥 ∩ ((int‘𝐽)‘𝐴))) ⊆ 𝑦)) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦))
4236, 41syl6 35 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → ((𝑥𝐽 ∧ (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
4342expdimp 453 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑥𝐽) → ((𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
4443rexlimdva 3152 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦)))
45 eleq2 2826 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑃𝑧𝑃𝑥))
46 imaeq2 6009 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
4746sseq1d 3975 . . . . . . . . . . 11 (𝑧 = 𝑥 → ((𝐹𝑧) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ 𝑦))
4845, 47anbi12d 631 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
4948cbvrexvw 3226 . . . . . . . . 9 (∃𝑧𝐽 (𝑃𝑧 ∧ (𝐹𝑧) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
5044, 49syl6ib 250 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
5118, 50impbid2 225 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
52 vex 3449 . . . . . . . . . 10 𝑥 ∈ V
5352inex1 5274 . . . . . . . . 9 (𝑥𝐴) ∈ V
5453a1i 11 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
5519uniexd 7679 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ V)
56 simp1r 1198 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐴𝑋)
5756, 5sseqtrdi 3994 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐴 𝐽)
5855, 57ssexd 5281 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐴 ∈ V)
59 elrest 17309 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
6019, 58, 59syl2anc 584 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
61 eleq2 2826 . . . . . . . . . 10 (𝑧 = (𝑥𝐴) → (𝑃𝑧𝑃 ∈ (𝑥𝐴)))
62 elin 3926 . . . . . . . . . . . 12 (𝑃 ∈ (𝑥𝐴) ↔ (𝑃𝑥𝑃𝐴))
6362rbaib 539 . . . . . . . . . . 11 (𝑃𝐴 → (𝑃 ∈ (𝑥𝐴) ↔ 𝑃𝑥))
649, 63syl 17 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝑃 ∈ (𝑥𝐴) ↔ 𝑃𝑥))
6561, 64sylan9bbr 511 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → (𝑃𝑧𝑃𝑥))
66 simpr 485 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → 𝑧 = (𝑥𝐴))
6766imaeq2d 6013 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = ((𝐹𝐴) “ (𝑥𝐴)))
68 inss2 4189 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝐴
69 resima2 5972 . . . . . . . . . . . 12 ((𝑥𝐴) ⊆ 𝐴 → ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴)))
7068, 69ax-mp 5 . . . . . . . . . . 11 ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴))
7167, 70eqtrdi 2792 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = (𝐹 “ (𝑥𝐴)))
7271sseq1d 3975 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → (((𝐹𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
7365, 72anbi12d 631 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) ∧ 𝑧 = (𝑥𝐴)) → ((𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
7454, 60, 73rexxfr2d 5366 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
7551, 74bitr4d 281 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) ↔ ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
7612, 75imbi12d 344 . . . . 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))))
7776ralbidv 3174 . . . 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))))
78 simp2r 1200 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐹:𝑋𝑌)
79 simp3 1138 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐾 ∈ Top)
8056, 9sseldd 3945 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝑃𝑋)
81 cnprest.2 . . . . . . . 8 𝑌 = 𝐾
825, 81iscnp2 22590 . . . . . . 7 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
8382baib 536 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
8419, 79, 80, 83syl3anc 1371 . . . . 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)))))
8578, 84mpbirand 705 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))))
8678, 56fssresd 6709 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹𝐴):𝐴𝑌)
875toptopon 22266 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
8819, 87sylib 217 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐽 ∈ (TopOn‘𝑋))
89 resttopon 22512 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
9088, 56, 89syl2anc 584 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
9181toptopon 22266 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
9279, 91sylib 217 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘𝑌))
93 iscnp 22588 . . . . . 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)))))
9490, 92, 9, 93syl3anc 1371 . . . . 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)))))
9586, 94mpbirand 705 . . . 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))))
9677, 85, 953bitr4d 310 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌) ∧ 𝐾 ∈ Top) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃)))
97963expia 1121 . 2 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐾 ∈ Top → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃))))
982, 4, 97pm5.21ndd 380 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  cin 3909  wss 3910   cuni 4865  cres 5635  cima 5636  wf 6492  cfv 6496  (class class class)co 7357  t crest 17302  Topctop 22242  TopOnctopon 22259  intcnt 22368   CnP ccnp 22576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-map 8767  df-en 8884  df-fin 8887  df-fi 9347  df-rest 17304  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-ntr 22371  df-cnp 22579
This theorem is referenced by:  limcres  25250  dvcnvrelem2  25382  psercn  25785  abelth  25800  cxpcn3  26101  efrlim  26319  cvmlift2lem11  33907  cvmlift2lem12  33908  cvmlift3lem7  33919  cncfuni  44117  cncfiooicclem1  44124  dirkercncflem4  44337  fourierdlem62  44399
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