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Theorem ovolficcss 24978
Description: Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)
Assertion
Ref Expression
ovolficcss (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ βˆͺ ran ([,] ∘ 𝐹) βŠ† ℝ)
Proof of Theorem ovolficcss
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnco2 6250 . . 3 ran ([,] ∘ 𝐹) = ([,] β€œ ran 𝐹)
2 ffvelcdm 7081 . . . . . . . . . . . 12 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ (πΉβ€˜π‘¦) ∈ ( ≀ ∩ (ℝ Γ— ℝ)))
32elin2d 4199 . . . . . . . . . . 11 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ (πΉβ€˜π‘¦) ∈ (ℝ Γ— ℝ))
4 1st2nd2 8011 . . . . . . . . . . 11 ((πΉβ€˜π‘¦) ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘¦) = ⟨(1st β€˜(πΉβ€˜π‘¦)), (2nd β€˜(πΉβ€˜π‘¦))⟩)
53, 4syl 17 . . . . . . . . . 10 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ (πΉβ€˜π‘¦) = ⟨(1st β€˜(πΉβ€˜π‘¦)), (2nd β€˜(πΉβ€˜π‘¦))⟩)
65fveq2d 6893 . . . . . . . . 9 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ ([,]β€˜(πΉβ€˜π‘¦)) = ([,]β€˜βŸ¨(1st β€˜(πΉβ€˜π‘¦)), (2nd β€˜(πΉβ€˜π‘¦))⟩))
7 df-ov 7409 . . . . . . . . 9 ((1st β€˜(πΉβ€˜π‘¦))[,](2nd β€˜(πΉβ€˜π‘¦))) = ([,]β€˜βŸ¨(1st β€˜(πΉβ€˜π‘¦)), (2nd β€˜(πΉβ€˜π‘¦))⟩)
86, 7eqtr4di 2791 . . . . . . . 8 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ ([,]β€˜(πΉβ€˜π‘¦)) = ((1st β€˜(πΉβ€˜π‘¦))[,](2nd β€˜(πΉβ€˜π‘¦))))
9 xp1st 8004 . . . . . . . . . 10 ((πΉβ€˜π‘¦) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΉβ€˜π‘¦)) ∈ ℝ)
103, 9syl 17 . . . . . . . . 9 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ (1st β€˜(πΉβ€˜π‘¦)) ∈ ℝ)
11 xp2nd 8005 . . . . . . . . . 10 ((πΉβ€˜π‘¦) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΉβ€˜π‘¦)) ∈ ℝ)
123, 11syl 17 . . . . . . . . 9 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ (2nd β€˜(πΉβ€˜π‘¦)) ∈ ℝ)
13 iccssre 13403 . . . . . . . . 9 (((1st β€˜(πΉβ€˜π‘¦)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘¦)) ∈ ℝ) β†’ ((1st β€˜(πΉβ€˜π‘¦))[,](2nd β€˜(πΉβ€˜π‘¦))) βŠ† ℝ)
1410, 12, 13syl2anc 585 . . . . . . . 8 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ ((1st β€˜(πΉβ€˜π‘¦))[,](2nd β€˜(πΉβ€˜π‘¦))) βŠ† ℝ)
158, 14eqsstrd 4020 . . . . . . 7 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ ([,]β€˜(πΉβ€˜π‘¦)) βŠ† ℝ)
16 reex 11198 . . . . . . . 8 ℝ ∈ V
1716elpw2 5345 . . . . . . 7 (([,]β€˜(πΉβ€˜π‘¦)) ∈ 𝒫 ℝ ↔ ([,]β€˜(πΉβ€˜π‘¦)) βŠ† ℝ)
1815, 17sylibr 233 . . . . . 6 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ ([,]β€˜(πΉβ€˜π‘¦)) ∈ 𝒫 ℝ)
1918ralrimiva 3147 . . . . 5 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ βˆ€π‘¦ ∈ β„• ([,]β€˜(πΉβ€˜π‘¦)) ∈ 𝒫 ℝ)
20 ffn 6715 . . . . . 6 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ 𝐹 Fn β„•)
21 fveq2 6889 . . . . . . . 8 (π‘₯ = (πΉβ€˜π‘¦) β†’ ([,]β€˜π‘₯) = ([,]β€˜(πΉβ€˜π‘¦)))
2221eleq1d 2819 . . . . . . 7 (π‘₯ = (πΉβ€˜π‘¦) β†’ (([,]β€˜π‘₯) ∈ 𝒫 ℝ ↔ ([,]β€˜(πΉβ€˜π‘¦)) ∈ 𝒫 ℝ))
2322ralrn 7087 . . . . . 6 (𝐹 Fn β„• β†’ (βˆ€π‘₯ ∈ ran 𝐹([,]β€˜π‘₯) ∈ 𝒫 ℝ ↔ βˆ€π‘¦ ∈ β„• ([,]β€˜(πΉβ€˜π‘¦)) ∈ 𝒫 ℝ))
2420, 23syl 17 . . . . 5 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ (βˆ€π‘₯ ∈ ran 𝐹([,]β€˜π‘₯) ∈ 𝒫 ℝ ↔ βˆ€π‘¦ ∈ β„• ([,]β€˜(πΉβ€˜π‘¦)) ∈ 𝒫 ℝ))
2519, 24mpbird 257 . . . 4 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ βˆ€π‘₯ ∈ ran 𝐹([,]β€˜π‘₯) ∈ 𝒫 ℝ)
26 iccf 13422 . . . . . 6 [,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
27 ffun 6718 . . . . . 6 ([,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ* β†’ Fun [,])
2826, 27ax-mp 5 . . . . 5 Fun [,]
29 frn 6722 . . . . . 6 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ ran 𝐹 βŠ† ( ≀ ∩ (ℝ Γ— ℝ)))
30 inss2 4229 . . . . . . . 8 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)
31 rexpssxrxp 11256 . . . . . . . 8 (ℝ Γ— ℝ) βŠ† (ℝ* Γ— ℝ*)
3230, 31sstri 3991 . . . . . . 7 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ* Γ— ℝ*)
3326fdmi 6727 . . . . . . 7 dom [,] = (ℝ* Γ— ℝ*)
3432, 33sseqtrri 4019 . . . . . 6 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† dom [,]
3529, 34sstrdi 3994 . . . . 5 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ ran 𝐹 βŠ† dom [,])
36 funimass4 6954 . . . . 5 ((Fun [,] ∧ ran 𝐹 βŠ† dom [,]) β†’ (([,] β€œ ran 𝐹) βŠ† 𝒫 ℝ ↔ βˆ€π‘₯ ∈ ran 𝐹([,]β€˜π‘₯) ∈ 𝒫 ℝ))
3728, 35, 36sylancr 588 . . . 4 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ (([,] β€œ ran 𝐹) βŠ† 𝒫 ℝ ↔ βˆ€π‘₯ ∈ ran 𝐹([,]β€˜π‘₯) ∈ 𝒫 ℝ))
3825, 37mpbird 257 . . 3 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ ([,] β€œ ran 𝐹) βŠ† 𝒫 ℝ)
391, 38eqsstrid 4030 . 2 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ ran ([,] ∘ 𝐹) βŠ† 𝒫 ℝ)
40 sspwuni 5103 . 2 (ran ([,] ∘ 𝐹) βŠ† 𝒫 ℝ ↔ βˆͺ ran ([,] ∘ 𝐹) βŠ† ℝ)
4139, 40sylib 217 1 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ βˆͺ ran ([,] ∘ 𝐹) βŠ† ℝ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  βŸ¨cop 4634  βˆͺ cuni 4908   Γ— cxp 5674  dom cdm 5676  ran crn 5677   β€œ cima 5679   ∘ ccom 5680  Fun wfun 6535   Fn wfn 6536  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406  1st c1st 7970  2nd c2nd 7971  β„cr 11106  β„*cxr 11244   ≀ cle 11246  β„•cn 12209  [,]cicc 13324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-pre-lttri 11181  ax-pre-lttrn 11182
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-icc 13328
This theorem is referenced by:  ovollb2lem  24997  ovollb2  24998  uniiccdif  25087  uniiccvol  25089  uniioombllem3  25094  uniioombllem4  25095  uniioombllem5  25096  uniiccmbl  25099
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