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Theorem ovolficcss 25385
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 6251 . . 3 ran ([,] ∘ 𝐹) = ([,] β€œ ran 𝐹)
2 ffvelcdm 7085 . . . . . . . . . . . 12 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ (πΉβ€˜π‘¦) ∈ ( ≀ ∩ (ℝ Γ— ℝ)))
32elin2d 4195 . . . . . . . . . . 11 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ (πΉβ€˜π‘¦) ∈ (ℝ Γ— ℝ))
4 1st2nd2 8026 . . . . . . . . . . 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 6895 . . . . . . . . 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 7417 . . . . . . . . 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 2785 . . . . . . . 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 8019 . . . . . . . . . 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 8020 . . . . . . . . . 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 13430 . . . . . . . . 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 583 . . . . . . . 8 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ ((1st β€˜(πΉβ€˜π‘¦))[,](2nd β€˜(πΉβ€˜π‘¦))) βŠ† ℝ)
158, 14eqsstrd 4016 . . . . . . 7 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑦 ∈ β„•) β†’ ([,]β€˜(πΉβ€˜π‘¦)) βŠ† ℝ)
16 reex 11221 . . . . . . . 8 ℝ ∈ V
1716elpw2 5341 . . . . . . 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 3141 . . . . 5 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ βˆ€π‘¦ ∈ β„• ([,]β€˜(πΉβ€˜π‘¦)) ∈ 𝒫 ℝ)
20 ffn 6716 . . . . . 6 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ 𝐹 Fn β„•)
21 fveq2 6891 . . . . . . . 8 (π‘₯ = (πΉβ€˜π‘¦) β†’ ([,]β€˜π‘₯) = ([,]β€˜(πΉβ€˜π‘¦)))
2221eleq1d 2813 . . . . . . 7 (π‘₯ = (πΉβ€˜π‘¦) β†’ (([,]β€˜π‘₯) ∈ 𝒫 ℝ ↔ ([,]β€˜(πΉβ€˜π‘¦)) ∈ 𝒫 ℝ))
2322ralrn 7092 . . . . . 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 13449 . . . . . 6 [,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
27 ffun 6719 . . . . . 6 ([,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ* β†’ Fun [,])
2826, 27ax-mp 5 . . . . 5 Fun [,]
29 frn 6723 . . . . . 6 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ ran 𝐹 βŠ† ( ≀ ∩ (ℝ Γ— ℝ)))
30 inss2 4225 . . . . . . . 8 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)
31 rexpssxrxp 11281 . . . . . . . 8 (ℝ Γ— ℝ) βŠ† (ℝ* Γ— ℝ*)
3230, 31sstri 3987 . . . . . . 7 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ* Γ— ℝ*)
3326fdmi 6728 . . . . . . 7 dom [,] = (ℝ* Γ— ℝ*)
3432, 33sseqtrri 4015 . . . . . 6 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† dom [,]
3529, 34sstrdi 3990 . . . . 5 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ ran 𝐹 βŠ† dom [,])
36 funimass4 6957 . . . . 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 586 . . . 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 4026 . 2 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ ran ([,] ∘ 𝐹) βŠ† 𝒫 ℝ)
40 sspwuni 5097 . 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 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056   ∩ cin 3943   βŠ† wss 3944  π’« cpw 4598  βŸ¨cop 4630  βˆͺ cuni 4903   Γ— cxp 5670  dom cdm 5672  ran crn 5673   β€œ cima 5675   ∘ ccom 5676  Fun wfun 6536   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414  1st c1st 7985  2nd c2nd 7986  β„cr 11129  β„*cxr 11269   ≀ cle 11271  β„•cn 12234  [,]cicc 13351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-pre-lttri 11204  ax-pre-lttrn 11205
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-po 5584  df-so 5585  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-icc 13355
This theorem is referenced by:  ovollb2lem  25404  ovollb2  25405  uniiccdif  25494  uniiccvol  25496  uniioombllem3  25501  uniioombllem4  25502  uniioombllem5  25503  uniiccmbl  25506
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