Theorem measfn 34506

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 34505	. 2 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
2	1	ffnd 6696	1 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀 Fn 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2145   Fn wfn 6520  ‘cfv 6525  (class class class)co 7400  0cc0 11088  +∞cpnf 11228  [,]cicc 13363  measurescmeas 34497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-ov 7403  df-esum 34330  df-meas 34498

This theorem is referenced by:  probfinmeasb  34730  dstfrvclim1  34780