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Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolficc Structured version   Visualization version   GIF version
Theorem ovolficc 25217
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 13429 . . . . . 6 [,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
2 inss2 4228 . . . . . . . 8 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)
3 rexpssxrxp 11263 . . . . . . . 8 (ℝ Γ— ℝ) βŠ† (ℝ* Γ— ℝ*)
42, 3sstri 3990 . . . . . . 7 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ* Γ— ℝ*)
5 fss 6733 . . . . . . 7 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ* Γ— ℝ*)) β†’ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*))
64, 5mpan2 687 . . . . . 6 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*))
7 fco 6740 . . . . . 6 (([,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ* ∧ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*)) β†’ ([,] ∘ 𝐹):β„•βŸΆπ’« ℝ*)
81, 6, 7sylancr 585 . . . . 5 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ ([,] ∘ 𝐹):β„•βŸΆπ’« ℝ*)
9 ffn 6716 . . . . 5 (([,] ∘ 𝐹):β„•βŸΆπ’« ℝ* β†’ ([,] ∘ 𝐹) Fn β„•)
10 fniunfv 7248 . . . . 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 4013 . . 3 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ (𝐴 βŠ† βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ 𝐴 βŠ† βˆͺ ran ([,] ∘ 𝐹)))
1312adantl 480 . 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 3969 . . 3 (𝐴 βŠ† βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ βˆ€π‘§ ∈ 𝐴 𝑧 ∈ βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›))
15 ssel2 3976 . . . . . 6 ((𝐴 βŠ† ℝ ∧ 𝑧 ∈ 𝐴) β†’ 𝑧 ∈ ℝ)
16 eliun 5000 . . . . . . 7 (𝑧 ∈ βˆͺ 𝑛 ∈ β„• (([,] ∘ 𝐹)β€˜π‘›) ↔ βˆƒπ‘› ∈ β„• 𝑧 ∈ (([,] ∘ 𝐹)β€˜π‘›))
17 simpll 763 . . . . . . . . 9 (((𝑧 ∈ ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑛 ∈ β„•) β†’ 𝑧 ∈ ℝ)
18 fvco3 6989 . . . . . . . . . . . . 13 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (([,] ∘ 𝐹)β€˜π‘›) = ([,]β€˜(πΉβ€˜π‘›)))
19 ffvelcdm 7082 . . . . . . . . . . . . . . . . 17 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ ( ≀ ∩ (ℝ Γ— ℝ)))
2019elin2d 4198 . . . . . . . . . . . . . . . 16 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ (ℝ Γ— ℝ))
21 1st2nd2 8016 . . . . . . . . . . . . . . . 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 6894 . . . . . . . . . . . . . 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 7414 . . . . . . . . . . . . . 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 2788 . . . . . . . . . . . . 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 2770 . . . . . . . . . . . 12 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (([,] ∘ 𝐹)β€˜π‘›) = ((1st β€˜(πΉβ€˜π‘›))[,](2nd β€˜(πΉβ€˜π‘›))))
2726eleq2d 2817 . . . . . . . . . . 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 25215 . . . . . . . . . . . 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 13393 . . . . . . . . . . . . . 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 1093 . . . . . . . . . . . . . 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 1130 . . . . . . . . . . . 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 710 . . . . . . . . 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 703 . . . . . . . 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 3174 . . . . . . 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 578 . . . . 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 648 . . . 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 3173 . . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  βŸ¨cop 4633  βˆͺ cuni 4907  βˆͺ ciun 4996   class class class wbr 5147   Γ— cxp 5673  ran crn 5676   ∘ ccom 5679   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  β„cr 11111  β„*cxr 11251   ≀ cle 11253  β„•cn 12216  [,]cicc 13331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-pre-lttri 11186  ax-pre-lttrn 11187
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-icc 13335
This theorem is referenced by:  ovollb2lem  25237  ovolctb  25239  ovolicc1  25265  ioombl1lem4  25310
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