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Theorem cnptopresti 14135
Description: One direction of cnptoprest 14136 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 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (𝐹 β†Ύ 𝐴) ∈ (((𝐽 β†Ύt 𝐴) CnP 𝐾)β€˜π‘ƒ))

Proof of Theorem cnptopresti
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 527 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 toptopon2 13916 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
32biimpi 120 . . . . 5 (𝐾 ∈ Top β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
43ad2antlr 489 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
5 simpr3 1007 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))
6 cnpf2 14104 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
71, 4, 5, 6syl3anc 1249 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
8 simpr1 1005 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ 𝐴 βŠ† 𝑋)
97, 8fssresd 5407 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (𝐹 β†Ύ 𝐴):𝐴⟢βˆͺ 𝐾)
10 simplr2 1042 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) β†’ 𝑃 ∈ 𝐴)
11 fvres 5554 . . . . . 6 (𝑃 ∈ 𝐴 β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) = (πΉβ€˜π‘ƒ))
1210, 11syl 14 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) = (πΉβ€˜π‘ƒ))
1312eleq1d 2258 . . . 4 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) β†’ (((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) ∈ 𝑦 ↔ (πΉβ€˜π‘ƒ) ∈ 𝑦))
141ad2antrr 488 . . . . . . 7 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
154ad2antrr 488 . . . . . . 7 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
168ad2antrr 488 . . . . . . . 8 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ 𝐴 βŠ† 𝑋)
17 simpr2 1006 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ 𝑃 ∈ 𝐴)
1817ad2antrr 488 . . . . . . . 8 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ 𝑃 ∈ 𝐴)
1916, 18sseldd 3171 . . . . . . 7 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ 𝑃 ∈ 𝑋)
205ad2antrr 488 . . . . . . 7 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))
21 simplr 528 . . . . . . 7 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ 𝑦 ∈ 𝐾)
22 simpr 110 . . . . . . 7 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ (πΉβ€˜π‘ƒ) ∈ 𝑦)
23 icnpimaex 14108 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝑃 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝑦 ∈ 𝐾 ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦)) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦))
2414, 15, 19, 20, 21, 22, 23syl33anc 1264 . . . . . 6 (((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) ∧ (πΉβ€˜π‘ƒ) ∈ 𝑦) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦))
2524ex 115 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) β†’ ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦)))
26 idd 21 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (𝑃 ∈ π‘₯ β†’ 𝑃 ∈ π‘₯))
2726, 17jctird 317 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (𝑃 ∈ π‘₯ β†’ (𝑃 ∈ π‘₯ ∧ 𝑃 ∈ 𝐴)))
28 elin 3333 . . . . . . . . . 10 (𝑃 ∈ (π‘₯ ∩ 𝐴) ↔ (𝑃 ∈ π‘₯ ∧ 𝑃 ∈ 𝐴))
2927, 28imbitrrdi 162 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (𝑃 ∈ π‘₯ β†’ 𝑃 ∈ (π‘₯ ∩ 𝐴)))
30 inss1 3370 . . . . . . . . . . . 12 (π‘₯ ∩ 𝐴) βŠ† π‘₯
31 imass2 5019 . . . . . . . . . . . 12 ((π‘₯ ∩ 𝐴) βŠ† π‘₯ β†’ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† (𝐹 β€œ π‘₯))
3230, 31ax-mp 5 . . . . . . . . . . 11 (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† (𝐹 β€œ π‘₯)
33 id 19 . . . . . . . . . . 11 ((𝐹 β€œ π‘₯) βŠ† 𝑦 β†’ (𝐹 β€œ π‘₯) βŠ† 𝑦)
3432, 33sstrid 3181 . . . . . . . . . 10 ((𝐹 β€œ π‘₯) βŠ† 𝑦 β†’ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦)
3534a1i 9 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ ((𝐹 β€œ π‘₯) βŠ† 𝑦 β†’ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦))
3629, 35anim12d 335 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ ((𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ (𝑃 ∈ (π‘₯ ∩ 𝐴) ∧ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦)))
3736reximdv 2591 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ (π‘₯ ∩ 𝐴) ∧ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦)))
38 vex 2755 . . . . . . . . . 10 π‘₯ ∈ V
3938inex1 4152 . . . . . . . . 9 (π‘₯ ∩ 𝐴) ∈ V
4039a1i 9 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ π‘₯ ∈ 𝐽) β†’ (π‘₯ ∩ 𝐴) ∈ V)
41 topontop 13911 . . . . . . . . . 10 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
4241ad2antrr 488 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ 𝐽 ∈ Top)
43 uniexg 4454 . . . . . . . . . . 11 (𝐽 ∈ Top β†’ βˆͺ 𝐽 ∈ V)
4442, 43syl 14 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ βˆͺ 𝐽 ∈ V)
45 toponuni 13912 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
4645sseq2d 3200 . . . . . . . . . . . 12 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐴 βŠ† 𝑋 ↔ 𝐴 βŠ† βˆͺ 𝐽))
4746ad2antrr 488 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (𝐴 βŠ† 𝑋 ↔ 𝐴 βŠ† βˆͺ 𝐽))
488, 47mpbid 147 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ 𝐴 βŠ† βˆͺ 𝐽)
4944, 48ssexd 4158 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ 𝐴 ∈ V)
50 elrest 12717 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) β†’ (𝑧 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐽 𝑧 = (π‘₯ ∩ 𝐴)))
5142, 49, 50syl2anc 411 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (𝑧 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐽 𝑧 = (π‘₯ ∩ 𝐴)))
52 simpr 110 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ 𝑧 = (π‘₯ ∩ 𝐴))
5352eleq2d 2259 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ (π‘₯ ∩ 𝐴)))
5452imaeq2d 4985 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) = ((𝐹 β†Ύ 𝐴) β€œ (π‘₯ ∩ 𝐴)))
55 inss2 3371 . . . . . . . . . . . 12 (π‘₯ ∩ 𝐴) βŠ† 𝐴
56 resima2 4956 . . . . . . . . . . . 12 ((π‘₯ ∩ 𝐴) βŠ† 𝐴 β†’ ((𝐹 β†Ύ 𝐴) β€œ (π‘₯ ∩ 𝐴)) = (𝐹 β€œ (π‘₯ ∩ 𝐴)))
5755, 56ax-mp 5 . . . . . . . . . . 11 ((𝐹 β†Ύ 𝐴) β€œ (π‘₯ ∩ 𝐴)) = (𝐹 β€œ (π‘₯ ∩ 𝐴))
5854, 57eqtrdi 2238 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) = (𝐹 β€œ (π‘₯ ∩ 𝐴)))
5958sseq1d 3199 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ (((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦 ↔ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦))
6053, 59anbi12d 473 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑧 = (π‘₯ ∩ 𝐴)) β†’ ((𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦) ↔ (𝑃 ∈ (π‘₯ ∩ 𝐴) ∧ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦)))
6140, 51, 60rexxfr2d 4480 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦) ↔ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ (π‘₯ ∩ 𝐴) ∧ (𝐹 β€œ (π‘₯ ∩ 𝐴)) βŠ† 𝑦)))
6237, 61sylibrd 169 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))
6362adantr 276 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) β†’ (βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝐹 β€œ π‘₯) βŠ† 𝑦) β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))
6425, 63syld 45 . . . 4 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) β†’ ((πΉβ€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))
6513, 64sylbid 150 . . 3 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) ∧ 𝑦 ∈ 𝐾) β†’ (((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))
6665ralrimiva 2563 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ βˆ€π‘¦ ∈ 𝐾 (((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))
67 resttopon 14068 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄))
681, 8, 67syl2anc 411 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄))
69 iscnp 14096 . . 3 (((𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄) ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝑃 ∈ 𝐴) β†’ ((𝐹 β†Ύ 𝐴) ∈ (((𝐽 β†Ύt 𝐴) CnP 𝐾)β€˜π‘ƒ) ↔ ((𝐹 β†Ύ 𝐴):𝐴⟢βˆͺ 𝐾 ∧ βˆ€π‘¦ ∈ 𝐾 (((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))))
7068, 4, 17, 69syl3anc 1249 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ ((𝐹 β†Ύ 𝐴) ∈ (((𝐽 β†Ύt 𝐴) CnP 𝐾)β€˜π‘ƒ) ↔ ((𝐹 β†Ύ 𝐴):𝐴⟢βˆͺ 𝐾 ∧ βˆ€π‘¦ ∈ 𝐾 (((𝐹 β†Ύ 𝐴)β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘§ ∈ (𝐽 β†Ύt 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 β†Ύ 𝐴) β€œ 𝑧) βŠ† 𝑦)))))
719, 66, 70mpbir2and 946 1 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ (𝐹 β†Ύ 𝐴) ∈ (((𝐽 β†Ύt 𝐴) CnP 𝐾)β€˜π‘ƒ))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  βˆ€wral 2468  βˆƒwrex 2469  Vcvv 2752   ∩ cin 3143   βŠ† wss 3144  βˆͺ cuni 3824   β†Ύ cres 4643   β€œ cima 4644  βŸΆwf 5227  β€˜cfv 5231  (class class class)co 5891   β†Ύt crest 12710  Topctop 13894  TopOnctopon 13907   CnP ccnp 14083
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-map 6668  df-rest 12712  df-topgen 12731  df-top 13895  df-topon 13908  df-bases 13940  df-cnp 14086
This theorem is referenced by: (None)
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