Theorem omsval 30355

Description: Value of the function mapping a content function to the corresponding outer measure. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)

Assertion

Ref	Expression
omsval	|- ( R e. _V -> (toOMeas ` R ) = ( a e. ~P U. dom R |-> inf ( ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) ) )

Distinct variable group:   x, a, y, z, R

Proof of Theorem omsval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oms 30354	. . 3 |- toOMeas = ( r e. _V |-> ( a e. ~P U. dom r |-> inf ( ran ( x e. { z e. ~P dom r | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( r ` y ) ) , ( 0 [,] +oo ) , < ) ) )
2	1	a1i 11	. 2 |- ( R e. _V -> toOMeas = ( r e. _V |-> ( a e. ~P U. dom r |-> inf ( ran ( x e. { z e. ~P dom r | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( r ` y ) ) , ( 0 [,] +oo ) , < ) ) ) )
3		dmeq 5324	. . . . . 6 |- ( r = R -> dom r = dom R )
4	3	unieqd 4446	. . . . 5 |- ( r = R -> U. dom r = U. dom R )
5	4	pweqd 4163	. . . 4 |- ( r = R -> ~P U. dom r = ~P U. dom R )
6	3	pweqd 4163	. . . . . . . 8 |- ( r = R -> ~P dom r = ~P dom R )
7		rabeq 3192	. . . . . . . 8 |- ( ~P dom r = ~P dom R -> { z e. ~P dom r | ( a C_ U. z /\ z ~<_ om ) } = { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } )
8	6, 7	syl 17	. . . . . . 7 |- ( r = R -> { z e. ~P dom r | ( a C_ U. z /\ z ~<_ om ) } = { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } )
9		simpl 473	. . . . . . . . 9 |- ( ( r = R /\ y e. x ) -> r = R )
10	9	fveq1d 6193	. . . . . . . 8 |- ( ( r = R /\ y e. x ) -> ( r ` y ) = ( R ` y ) )
11	10	esumeq2dv 30100	. . . . . . 7 |- ( r = R -> Σ* y e. x ( r ` y ) = Σ* y e. x ( R ` y ) )
12	8, 11	mpteq12dv 4733	. . . . . 6 |- ( r = R -> ( x e. { z e. ~P dom r | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( r ` y ) ) = ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) )
13	12	rneqd 5353	. . . . 5 |- ( r = R -> ran ( x e. { z e. ~P dom r | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( r ` y ) ) = ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) )
14	13	infeq1d 8383	. . . 4 |- ( r = R -> inf ( ran ( x e. { z e. ~P dom r | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( r ` y ) ) , ( 0 [,] +oo ) , < ) = inf ( ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) )
15	5, 14	mpteq12dv 4733	. . 3 |- ( r = R -> ( a e. ~P U. dom r |-> inf ( ran ( x e. { z e. ~P dom r | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( r ` y ) ) , ( 0 [,] +oo ) , < ) ) = ( a e. ~P U. dom R |-> inf ( ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) ) )
16	15	adantl 482	. 2 |- ( ( R e. _V /\ r = R ) -> ( a e. ~P U. dom r |-> inf ( ran ( x e. { z e. ~P dom r | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( r ` y ) ) , ( 0 [,] +oo ) , < ) ) = ( a e. ~P U. dom R |-> inf ( ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) ) )
17		id 22	. 2 |- ( R e. _V -> R e. _V )
18		dmexg 7097	. . 3 |- ( R e. _V -> dom R e. _V )
19		uniexg 6955	. . 3 |- ( dom R e. _V -> U. dom R e. _V )
20		pwexg 4850	. . 3 |- ( U. dom R e. _V -> ~P U. dom R e. _V )
21		mptexg 6484	. . 3 |- ( ~P U. dom R e. _V -> ( a e. ~P U. dom R |-> inf ( ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) ) e. _V )
22	18, 19, 20, 21	4syl 19	. 2 |- ( R e. _V -> ( a e. ~P U. dom R |-> inf ( ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) ) e. _V )
23	2, 16, 17, 22	fvmptd 6288	1 |- ( R e. _V -> (toOMeas ` R ) = ( a e. ~P U. dom R |-> inf ( ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } |-> Σ* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953  infcinf 8347   0cc0 9936   +oocpnf 10071   < clt 10074   [,]cicc 12178  Σ^*cesum 30089  toOMeascoms 30353

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-sup 8348  df-inf 8349  df-esum 30090  df-oms 30354

This theorem is referenced by:  omsfval  30356  omsf  30358