Theorem iccf 13432

Description: The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Assertion

Ref	Expression
iccf	⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*

Proof of Theorem iccf

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-icc 13338	. 2 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
2	1	ixxf 13341	1 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*

Syntax hints:  𝒫 cpw 4602   × cxp 5674  ⟶wf 6539  ℝ^*cxr 11254   ≤ cle 11256  [,]cicc 13334

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-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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-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-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-fv 6551  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-xr 11259  df-icc 13338

This theorem is referenced by:  lecldbas  22956  ovolficc  25230  ovolficcss  25231  uniiccdif  25340  uniiccvol  25342  dyadmbllem  25361  dyadmbl  25362  opnmbllem  25363  opnmbllem0  36840  mblfinlem1  36841  mblfinlem2  36842