Theorem measfn 34238

Description: A measure is a function on its base sigma-algebra. (Contributed by Thierry Arnoux, 13-Feb-2017.)

Assertion

Ref	Expression
measfn	⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀 Fn 𝑆)

Proof of Theorem measfn

Step	Hyp	Ref	Expression
1		measfrge0 34237	. 2 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
2	1	ffnd 6657	1 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀 Fn 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2113   Fn wfn 6481  ‘cfv 6486  (class class class)co 7352  0cc0 11013  +∞cpnf 11150  [,]cicc 13250  measurescmeas 34229

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7355  df-esum 34062  df-meas 34230

This theorem is referenced by:  probfinmeasb  34462  dstfrvclim1  34512