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Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolficc Structured version   Visualization version   GIF version
Theorem ovolficc 24884
Description: Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
Assertion
Ref Expression
ovolficc ((𝐴 βŠ† ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (𝐴 βŠ† βˆͺ ran ([,] ∘ 𝐹) ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘› ∈ β„• ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
Distinct variable groups:   𝑧,𝑛,𝐴   𝑛,𝐹,𝑧
Proof of Theorem ovolficc
StepHypRef Expression
1 iccf 13390 . . . . . 6 [,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
2 inss2 4209 . . . . . . . 8 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)
3 rexpssxrxp 11224 . . . . . . . 8 (ℝ Γ— ℝ) βŠ† (ℝ* Γ— ℝ*)
42, 3sstri 3971 . . . . . . 7 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ* Γ— ℝ*)
5 fss 6705 . . . . . . 7 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ* Γ— ℝ*)) β†’ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*))
64, 5mpan2 689 . . . . . 6 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*))
7 fco 6712 . . . . . 6 (([,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ* ∧ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*)) β†’ ([,] ∘ 𝐹):β„•βŸΆπ’« ℝ*)
81, 6, 7sylancr 587 . . . . 5 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ ([,] ∘ 𝐹):β„•βŸΆπ’« ℝ*)
9 ffn 6688 . . . . 5 (([,] ∘ 𝐹):β„•βŸΆπ’« ℝ* β†’ ([,] ∘ 𝐹) Fn β„•)
10 fniunfv 7214 . . . . 5 (([,] ∘ 𝐹) Fn β„• β†’ βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) = βˆͺ ran ([,] ∘ 𝐹))
118, 9, 103syl 18 . . . 4 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) = βˆͺ ran ([,] ∘ 𝐹))
1211sseq2d 3994 . . 3 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ (𝐴 βŠ† βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ 𝐴 βŠ† βˆͺ ran ([,] ∘ 𝐹)))
1312adantl 482 . 2 ((𝐴 βŠ† ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (𝐴 βŠ† βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ 𝐴 βŠ† βˆͺ ran ([,] ∘ 𝐹)))
14 dfss3 3950 . . 3 (𝐴 βŠ† βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ βˆ€π‘§ ∈ 𝐴 𝑧 ∈ βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›))
15 ssel2 3957 . . . . . 6 ((𝐴 βŠ† ℝ ∧ 𝑧 ∈ 𝐴) β†’ 𝑧 ∈ ℝ)
16 eliun 4978 . . . . . . 7 (𝑧 ∈ βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ βˆƒπ‘› ∈ β„• 𝑧 ∈ (([,] ∘ 𝐹)β€˜π‘›))
17 simpll 765 . . . . . . . . 9 (((𝑧 ∈ ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑛 ∈ β„•) β†’ 𝑧 ∈ ℝ)
18 fvco3 6960 . . . . . . . . . . . . 13 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (([,] ∘ 𝐹)β€˜π‘›) = ([,]β€˜(πΉβ€˜π‘›)))
19 ffvelcdm 7052 . . . . . . . . . . . . . . . . 17 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ ( ≀ ∩ (ℝ Γ— ℝ)))
2019elin2d 4179 . . . . . . . . . . . . . . . 16 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ (ℝ Γ— ℝ))
21 1st2nd2 7980 . . . . . . . . . . . . . . . 16 ((πΉβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘›) = ⟨(1st β€˜(πΉβ€˜π‘›)), (2nd β€˜(πΉβ€˜π‘›))⟩)
2220, 21syl 17 . . . . . . . . . . . . . . 15 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) = ⟨(1st β€˜(πΉβ€˜π‘›)), (2nd β€˜(πΉβ€˜π‘›))⟩)
2322fveq2d 6866 . . . . . . . . . . . . . 14 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ ([,]β€˜(πΉβ€˜π‘›)) = ([,]β€˜βŸ¨(1st β€˜(πΉβ€˜π‘›)), (2nd β€˜(πΉβ€˜π‘›))⟩))
24 df-ov 7380 . . . . . . . . . . . . . 14 ((1st β€˜(πΉβ€˜π‘›))[,](2nd β€˜(πΉβ€˜π‘›))) = ([,]β€˜βŸ¨(1st β€˜(πΉβ€˜π‘›)), (2nd β€˜(πΉβ€˜π‘›))⟩)
2523, 24eqtr4di 2789 . . . . . . . . . . . . 13 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ ([,]β€˜(πΉβ€˜π‘›)) = ((1st β€˜(πΉβ€˜π‘›))[,](2nd β€˜(πΉβ€˜π‘›))))
2618, 25eqtrd 2771 . . . . . . . . . . . 12 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (([,] ∘ 𝐹)β€˜π‘›) = ((1st β€˜(πΉβ€˜π‘›))[,](2nd β€˜(πΉβ€˜π‘›))))
2726eleq2d 2818 . . . . . . . . . . 11 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (𝑧 ∈ (([,] ∘ 𝐹)β€˜π‘›) ↔ 𝑧 ∈ ((1st β€˜(πΉβ€˜π‘›))[,](2nd β€˜(πΉβ€˜π‘›)))))
28 ovolfcl 24882 . . . . . . . . . . . 12 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ ((1st β€˜(πΉβ€˜π‘›)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘›)) ≀ (2nd β€˜(πΉβ€˜π‘›))))
29 elicc2 13354 . . . . . . . . . . . . . 14 (((1st β€˜(πΉβ€˜π‘›)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ) β†’ (𝑧 ∈ ((1st β€˜(πΉβ€˜π‘›))[,](2nd β€˜(πΉβ€˜π‘›))) ↔ (𝑧 ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
30 3anass 1095 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›))) ↔ (𝑧 ∈ ℝ ∧ ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
3129, 30bitrdi 286 . . . . . . . . . . . . 13 (((1st β€˜(πΉβ€˜π‘›)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ) β†’ (𝑧 ∈ ((1st β€˜(πΉβ€˜π‘›))[,](2nd β€˜(πΉβ€˜π‘›))) ↔ (𝑧 ∈ ℝ ∧ ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›))))))
32313adant3 1132 . . . . . . . . . . . 12 (((1st β€˜(πΉβ€˜π‘›)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘›)) ≀ (2nd β€˜(πΉβ€˜π‘›))) β†’ (𝑧 ∈ ((1st β€˜(πΉβ€˜π‘›))[,](2nd β€˜(πΉβ€˜π‘›))) ↔ (𝑧 ∈ ℝ ∧ ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›))))))
3328, 32syl 17 . . . . . . . . . . 11 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (𝑧 ∈ ((1st β€˜(πΉβ€˜π‘›))[,](2nd β€˜(πΉβ€˜π‘›))) ↔ (𝑧 ∈ ℝ ∧ ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›))))))
3427, 33bitrd 278 . . . . . . . . . 10 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (𝑧 ∈ (([,] ∘ 𝐹)β€˜π‘›) ↔ (𝑧 ∈ ℝ ∧ ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›))))))
3534adantll 712 . . . . . . . . 9 (((𝑧 ∈ ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑛 ∈ β„•) β†’ (𝑧 ∈ (([,] ∘ 𝐹)β€˜π‘›) ↔ (𝑧 ∈ ℝ ∧ ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›))))))
3617, 35mpbirand 705 . . . . . . . 8 (((𝑧 ∈ ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑛 ∈ β„•) β†’ (𝑧 ∈ (([,] ∘ 𝐹)β€˜π‘›) ↔ ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
3736rexbidva 3175 . . . . . . 7 ((𝑧 ∈ ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (βˆƒπ‘› ∈ β„• 𝑧 ∈ (([,] ∘ 𝐹)β€˜π‘›) ↔ βˆƒπ‘› ∈ β„• ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
3816, 37bitrid 282 . . . . . 6 ((𝑧 ∈ ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (𝑧 ∈ βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ βˆƒπ‘› ∈ β„• ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
3915, 38sylan 580 . . . . 5 (((𝐴 βŠ† ℝ ∧ 𝑧 ∈ 𝐴) ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (𝑧 ∈ βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ βˆƒπ‘› ∈ β„• ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
4039an32s 650 . . . 4 (((𝐴 βŠ† ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑧 ∈ 𝐴) β†’ (𝑧 ∈ βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ βˆƒπ‘› ∈ β„• ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
4140ralbidva 3174 . . 3 ((𝐴 βŠ† ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (βˆ€π‘§ ∈ 𝐴 𝑧 ∈ βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘› ∈ β„• ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
4214, 41bitrid 282 . 2 ((𝐴 βŠ† ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (𝐴 βŠ† βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘› ∈ β„• ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
4313, 42bitr3d 280 1 ((𝐴 βŠ† ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (𝐴 βŠ† βˆͺ ran ([,] ∘ 𝐹) ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘› ∈ β„• ((1st β€˜(πΉβ€˜π‘›)) ≀ 𝑧 ∧ 𝑧 ≀ (2nd β€˜(πΉβ€˜π‘›)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3060  βˆƒwrex 3069   ∩ cin 3927   βŠ† wss 3928  π’« cpw 4580  βŸ¨cop 4612  βˆͺ cuni 4885  βˆͺ ciun 4974   class class class wbr 5125   Γ— cxp 5651  ran crn 5654   ∘ ccom 5657   Fn wfn 6511  βŸΆwf 6512  β€˜cfv 6516  (class class class)co 7377  1st c1st 7939  2nd c2nd 7940  β„cr 11074  β„*cxr 11212   ≀ cle 11214  β„•cn 12177  [,]cicc 13292
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 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-pre-lttri 11149  ax-pre-lttrn 11150
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-po 5565  df-so 5566  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-1st 7941  df-2nd 7942  df-er 8670  df-en 8906  df-dom 8907  df-sdom 8908  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-icc 13296
This theorem is referenced by:  ovollb2lem  24904  ovolctb  24906  ovolicc1  24932  ioombl1lem4  24977
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