Theorem elprob 30471

Description: The property of being a probability measure. (Contributed by Thierry Arnoux, 8-Dec-2016.)

Assertion

Ref	Expression
elprob	|- ( P e. Prob <-> ( P e. U. ran measures /\ ( P ` U. dom P ) = 1 ) )

Proof of Theorem elprob

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( p = P -> p = P )
2		dmeq 5324	. . . . 5 |- ( p = P -> dom p = dom P )
3	2	unieqd 4446	. . . 4 |- ( p = P -> U. dom p = U. dom P )
4	1, 3	fveq12d 6197	. . 3 |- ( p = P -> ( p ` U. dom p ) = ( P ` U. dom P ) )
5	4	eqeq1d 2624	. 2 |- ( p = P -> ( ( p ` U. dom p ) = 1 <-> ( P ` U. dom P ) = 1 ) )
6		df-prob 30470	. 2 |- Prob = { p e. U. ran measures | ( p ` U. dom p ) = 1 }
7	5, 6	elrab2 3366	1 |- ( P e. Prob <-> ( P e. U. ran measures /\ ( P ` U. dom P ) = 1 ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   U.cuni 4436   dom cdm 5114   ran crn 5115   ` cfv 5888   1c1 9937  measurescmeas 30258  Probcprb 30469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896  df-prob 30470

This theorem is referenced by:  domprobmeas  30472  probtot  30474  probfinmeasbOLD  30490  probfinmeasb  30491  probmeasb  30492  dstrvprob  30533