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Theorem cnptopresti 13231
Description: One direction of cnptoprest 13232 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 13010 . . . . . 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 1005 . . . 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 13200 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ)) β†’ 𝐹:π‘‹βŸΆβˆͺ 𝐾)
71, 4, 5, 6syl3anc 1238 . . 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 1003 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ 𝐴 βŠ† 𝑋)
97, 8fssresd 5384 . 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 1040 . . . . . 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 5531 . . . . . 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 2244 . . . 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 1004 . . . . . . . . 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 3154 . . . . . . 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 13204 . . . . . . 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 1253 . . . . . 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 3316 . . . . . . . . . 10 (𝑃 ∈ (π‘₯ ∩ 𝐴) ↔ (𝑃 ∈ π‘₯ ∧ 𝑃 ∈ 𝐴))
2927, 28syl6ibr 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 3353 . . . . . . . . . . . 12 (π‘₯ ∩ 𝐴) βŠ† π‘₯
31 imass2 4997 . . . . . . . . . . . 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 3164 . . . . . . . . . 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 2576 . . . . . . 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 2738 . . . . . . . . . 10 π‘₯ ∈ V
3938inex1 4132 . . . . . . . . 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 13005 . . . . . . . . . 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 4433 . . . . . . . . . . 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 13006 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
4645sseq2d 3183 . . . . . . . . . . . 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 4138 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Top) ∧ (𝐴 βŠ† 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ))) β†’ 𝐴 ∈ V)
50 elrest 12616 . . . . . . . . 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 2245 . . . . . . . . 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 4963 . . . . . . . . . . 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 3354 . . . . . . . . . . . 12 (π‘₯ ∩ 𝐴) βŠ† 𝐴
56 resima2 4934 . . . . . . . . . . . 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 2224 . . . . . . . . . 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 3182 . . . . . . . . 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 4459 . . . . . . 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 2548 . 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 13164 . . . 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 13192 . . 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 1238 . 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 944 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 978   = wceq 1353   ∈ wcel 2146  βˆ€wral 2453  βˆƒwrex 2454  Vcvv 2735   ∩ cin 3126   βŠ† wss 3127  βˆͺ cuni 3805   β†Ύ cres 4622   β€œ cima 4623  βŸΆwf 5204  β€˜cfv 5208  (class class class)co 5865   β†Ύt crest 12609  Topctop 12988  TopOnctopon 13001   CnP ccnp 13179
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-map 6640  df-rest 12611  df-topgen 12630  df-top 12989  df-topon 13002  df-bases 13034  df-cnp 13182
This theorem is referenced by: (None)
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