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Theorem cnptopresti 12421
 Description: One direction of cnptoprest 12422 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 31-Mar-2023.)
Assertion
Ref Expression
cnptopresti (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃))

Proof of Theorem cnptopresti
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 518 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐽 ∈ (TopOn‘𝑋))
2 toptopon2 12200 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
32biimpi 119 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘ 𝐾))
43ad2antlr 480 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐾 ∈ (TopOn‘ 𝐾))
5 simpr3 989 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
6 cnpf2 12390 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋 𝐾)
71, 4, 5, 6syl3anc 1216 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐹:𝑋 𝐾)
8 simpr1 987 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐴𝑋)
97, 8fssresd 5299 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝐹𝐴):𝐴 𝐾)
10 simplr2 1024 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) → 𝑃𝐴)
11 fvres 5445 . . . . . 6 (𝑃𝐴 → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
1210, 11syl 14 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) → ((𝐹𝐴)‘𝑃) = (𝐹𝑃))
1312eleq1d 2208 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) → (((𝐹𝐴)‘𝑃) ∈ 𝑦 ↔ (𝐹𝑃) ∈ 𝑦))
141ad2antrr 479 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) ∧ (𝐹𝑃) ∈ 𝑦) → 𝐽 ∈ (TopOn‘𝑋))
154ad2antrr 479 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) ∧ (𝐹𝑃) ∈ 𝑦) → 𝐾 ∈ (TopOn‘ 𝐾))
168ad2antrr 479 . . . . . . . 8 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) ∧ (𝐹𝑃) ∈ 𝑦) → 𝐴𝑋)
17 simpr2 988 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝑃𝐴)
1817ad2antrr 479 . . . . . . . 8 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) ∧ (𝐹𝑃) ∈ 𝑦) → 𝑃𝐴)
1916, 18sseldd 3098 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) ∧ (𝐹𝑃) ∈ 𝑦) → 𝑃𝑋)
205ad2antrr 479 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) ∧ (𝐹𝑃) ∈ 𝑦) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
21 simplr 519 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) ∧ (𝐹𝑃) ∈ 𝑦) → 𝑦𝐾)
22 simpr 109 . . . . . . 7 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) ∧ (𝐹𝑃) ∈ 𝑦) → (𝐹𝑃) ∈ 𝑦)
23 icnpimaex 12394 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑃𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦)) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
2414, 15, 19, 20, 21, 22, 23syl33anc 1231 . . . . . 6 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) ∧ (𝐹𝑃) ∈ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
2524ex 114 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) → ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
26 idd 21 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝑃𝑥𝑃𝑥))
2726, 17jctird 315 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝑃𝑥 → (𝑃𝑥𝑃𝐴)))
28 elin 3259 . . . . . . . . . 10 (𝑃 ∈ (𝑥𝐴) ↔ (𝑃𝑥𝑃𝐴))
2927, 28syl6ibr 161 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝑃𝑥𝑃 ∈ (𝑥𝐴)))
30 inss1 3296 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝑥
31 imass2 4915 . . . . . . . . . . . 12 ((𝑥𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥))
3230, 31ax-mp 5 . . . . . . . . . . 11 (𝐹 “ (𝑥𝐴)) ⊆ (𝐹𝑥)
33 id 19 . . . . . . . . . . 11 ((𝐹𝑥) ⊆ 𝑦 → (𝐹𝑥) ⊆ 𝑦)
3432, 33sstrid 3108 . . . . . . . . . 10 ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)
3534a1i 9 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → ((𝐹𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
3629, 35anim12d 333 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
3736reximdv 2533 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
38 vex 2689 . . . . . . . . . 10 𝑥 ∈ V
3938inex1 4062 . . . . . . . . 9 (𝑥𝐴) ∈ V
4039a1i 9 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
41 topontop 12195 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4241ad2antrr 479 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐽 ∈ Top)
43 uniexg 4361 . . . . . . . . . . 11 (𝐽 ∈ Top → 𝐽 ∈ V)
4442, 43syl 14 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐽 ∈ V)
45 toponuni 12196 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
4645sseq2d 3127 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → (𝐴𝑋𝐴 𝐽))
4746ad2antrr 479 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝐴𝑋𝐴 𝐽))
488, 47mpbid 146 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐴 𝐽)
4944, 48ssexd 4068 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → 𝐴 ∈ V)
50 elrest 12141 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
5142, 49, 50syl2anc 408 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝑧 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑧 = (𝑥𝐴)))
52 simpr 109 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥𝐴)) → 𝑧 = (𝑥𝐴))
5352eleq2d 2209 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥𝐴)) → (𝑃𝑧𝑃 ∈ (𝑥𝐴)))
5452imaeq2d 4881 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = ((𝐹𝐴) “ (𝑥𝐴)))
55 inss2 3297 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝐴
56 resima2 4853 . . . . . . . . . . . 12 ((𝑥𝐴) ⊆ 𝐴 → ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴)))
5755, 56ax-mp 5 . . . . . . . . . . 11 ((𝐹𝐴) “ (𝑥𝐴)) = (𝐹 “ (𝑥𝐴))
5854, 57syl6eq 2188 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥𝐴)) → ((𝐹𝐴) “ 𝑧) = (𝐹 “ (𝑥𝐴)))
5958sseq1d 3126 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥𝐴)) → (((𝐹𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦))
6053, 59anbi12d 464 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑧 = (𝑥𝐴)) → ((𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
6140, 51, 60rexxfr2d 4386 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃 ∈ (𝑥𝐴) ∧ (𝐹 “ (𝑥𝐴)) ⊆ 𝑦)))
6237, 61sylibrd 168 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
6362adantr 274 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
6425, 63syld 45 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) → ((𝐹𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
6513, 64sylbid 149 . . 3 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) ∧ 𝑦𝐾) → (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
6665ralrimiva 2505 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))
67 resttopon 12354 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
681, 8, 67syl2anc 408 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
69 iscnp 12382 . . 3 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑃𝐴) → ((𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹𝐴):𝐴 𝐾 ∧ ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))))
7068, 4, 17, 69syl3anc 1216 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))) → ((𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹𝐴):𝐴 𝐾 ∧ ∀𝑦𝐾 (((𝐹𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽t 𝐴)(𝑃𝑧 ∧ ((𝐹𝐴) “ 𝑧) ⊆ 𝑦)))))
719, 66, 70mpbir2and 928 1 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝑃𝐴𝐹 ∈ ((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 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  Vcvv 2686   ∩ cin 3070   ⊆ wss 3071  ∪ cuni 3736   ↾ cres 4541   “ cima 4542  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774   ↾t crest 12134  Topctop 12178  TopOnctopon 12191   CnP ccnp 12369 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-rest 12136  df-topgen 12155  df-top 12179  df-topon 12192  df-bases 12224  df-cnp 12372 This theorem is referenced by: (None)
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