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Theorem ovolfioo 25217
Description: Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)
Assertion
Ref Expression
ovolfioo ((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 ovolfioo
StepHypRef Expression
1 ioof 13429 . . . . . 6 (,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ
2 inss2 4229 . . . . . . . 8 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)
3 rexpssxrxp 11264 . . . . . . . 8 (ℝ Γ— ℝ) βŠ† (ℝ* Γ— ℝ*)
42, 3sstri 3991 . . . . . . 7 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ* Γ— ℝ*)
5 fss 6734 . . . . . . 7 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ* Γ— ℝ*)) β†’ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*))
64, 5mpan2 688 . . . . . 6 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*))
7 fco 6741 . . . . . 6 (((,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ ∧ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*)) β†’ ((,) ∘ 𝐹):β„•βŸΆπ’« ℝ)
81, 6, 7sylancr 586 . . . . 5 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ ((,) ∘ 𝐹):β„•βŸΆπ’« ℝ)
9 ffn 6717 . . . . 5 (((,) ∘ 𝐹):β„•βŸΆπ’« ℝ β†’ ((,) ∘ 𝐹) Fn β„•)
10 fniunfv 7249 . . . . 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 4014 . . 3 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ (𝐴 βŠ† βˆͺ 𝑛 ∈ β„• (((,) ∘ 𝐹)β€˜π‘›) ↔ 𝐴 βŠ† βˆͺ ran ((,) ∘ 𝐹)))
1312adantl 481 . 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 3970 . . 3 (𝐴 βŠ† βˆͺ 𝑛 ∈ β„• (((,) ∘ 𝐹)β€˜π‘›) ↔ βˆ€π‘§ ∈ 𝐴 𝑧 ∈ βˆͺ 𝑛 ∈ β„• (((,) ∘ 𝐹)β€˜π‘›))
15 ssel2 3977 . . . . . 6 ((𝐴 βŠ† ℝ ∧ 𝑧 ∈ 𝐴) β†’ 𝑧 ∈ ℝ)
16 eliun 5001 . . . . . . 7 (𝑧 ∈ βˆͺ 𝑛 ∈ β„• (((,) ∘ 𝐹)β€˜π‘›) ↔ βˆƒπ‘› ∈ β„• 𝑧 ∈ (((,) ∘ 𝐹)β€˜π‘›))
17 rexr 11265 . . . . . . . . . 10 (𝑧 ∈ ℝ β†’ 𝑧 ∈ ℝ*)
1817ad2antrr 723 . . . . . . . . 9 (((𝑧 ∈ ℝ ∧ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) ∧ 𝑛 ∈ β„•) β†’ 𝑧 ∈ ℝ*)
19 fvco3 6990 . . . . . . . . . . . . 13 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (((,) ∘ 𝐹)β€˜π‘›) = ((,)β€˜(πΉβ€˜π‘›)))
20 ffvelcdm 7083 . . . . . . . . . . . . . . . . 17 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ ( ≀ ∩ (ℝ Γ— ℝ)))
2120elin2d 4199 . . . . . . . . . . . . . . . 16 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ (ℝ Γ— ℝ))
22 1st2nd2 8017 . . . . . . . . . . . . . . . 16 ((πΉβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘›) = ⟨(1st β€˜(πΉβ€˜π‘›)), (2nd β€˜(πΉβ€˜π‘›))⟩)
2321, 22syl 17 . . . . . . . . . . . . . . 15 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) = ⟨(1st β€˜(πΉβ€˜π‘›)), (2nd β€˜(πΉβ€˜π‘›))⟩)
2423fveq2d 6895 . . . . . . . . . . . . . 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‘›))⟩))
25 df-ov 7415 . . . . . . . . . . . . . 14 ((1st β€˜(πΉβ€˜π‘›))(,)(2nd β€˜(πΉβ€˜π‘›))) = ((,)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘›)), (2nd β€˜(πΉβ€˜π‘›))⟩)
2624, 25eqtr4di 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‘›))))
2719, 26eqtrd 2771 . . . . . . . . . . . 12 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ (((,) ∘ 𝐹)β€˜π‘›) = ((1st β€˜(πΉβ€˜π‘›))(,)(2nd β€˜(πΉβ€˜π‘›))))
2827eleq2d 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‘›)))))
29 ovolfcl 25216 . . . . . . . . . . . 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‘›))))
30 rexr 11265 . . . . . . . . . . . . . . 15 ((1st β€˜(πΉβ€˜π‘›)) ∈ ℝ β†’ (1st β€˜(πΉβ€˜π‘›)) ∈ ℝ*)
31 rexr 11265 . . . . . . . . . . . . . . 15 ((2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ β†’ (2nd β€˜(πΉβ€˜π‘›)) ∈ ℝ*)
32 elioo1 13369 . . . . . . . . . . . . . . 15 (((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‘›)))))
3330, 31, 32syl2an 595 . . . . . . . . . . . . . 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‘›)))))
34 3anass 1094 . . . . . . . . . . . . . 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‘›)))))
3533, 34bitrdi 287 . . . . . . . . . . . . 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‘›))))))
36353adant3 1131 . . . . . . . . . . . 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‘›))))))
3729, 36syl 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‘›))))))
3828, 37bitrd 279 . . . . . . . . . 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‘›))))))
3938adantll 711 . . . . . . . . 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‘›))))))
4018, 39mpbirand 704 . . . . . . . 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‘›)))))
4140rexbidva 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‘›)))))
4216, 41bitrid 283 . . . . . 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‘›)))))
4315, 42sylan 579 . . . . 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‘›)))))
4443an32s 649 . . . 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‘›)))))
4544ralbidva 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‘›)))))
4614, 45bitrid 283 . 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‘›)))))
4713, 46bitr3d 281 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  βŸ¨cop 4634  βˆͺ cuni 4908  βˆͺ ciun 4997   class class class wbr 5148   Γ— cxp 5674  ran crn 5677   ∘ ccom 5680   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412  1st c1st 7976  2nd c2nd 7977  β„cr 11112  β„*cxr 11252   < clt 11253   ≀ cle 11254  β„•cn 12217  (,)cioo 13329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-pre-lttri 11187  ax-pre-lttrn 11188
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-ioo 13333
This theorem is referenced by:  ovollb2lem  25238  ovolunlem1  25247  ovoliunlem2  25253  ovolshftlem1  25259  ovolscalem1  25263  ioombl1lem4  25311
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