Theorem omsfval 30356

Description: Value of the outer measure evaluated for a given set A. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)

Assertion

Ref	Expression
omsfval	|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> ( (toOMeas ` R ) ` A ) = 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 groups:   x, y, z, R   x, A, y, z   x, Q, y, z   x, V, y, z

Proof of Theorem omsfval

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. . . 4 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> R : Q --> ( 0 [,] +oo ) )
2		simp1 1061	. . . 4 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> Q e. V )
3		fex 6490	. . . 4 |- ( ( R : Q --> ( 0 [,] +oo ) /\ Q e. V ) -> R e. _V )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> R e. _V )
5		omsval 30355	. . 3 |- ( 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 ) , < ) ) )
6	4, 5	syl 17	. 2 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> (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 ) , < ) ) )
7		simpr 477	. . . . . . . 8 |- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> a = A )
8	7	sseq1d 3632	. . . . . . 7 |- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> ( a C_ U. z <-> A C_ U. z ) )
9	8	anbi1d 741	. . . . . 6 |- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> ( ( a C_ U. z /\ z ~<_ om ) <-> ( A C_ U. z /\ z ~<_ om ) ) )
10	9	rabbidv 3189	. . . . 5 |- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> { z e. ~P dom R | ( a C_ U. z /\ z ~<_ om ) } = { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } )
11	10	mpteq1d 4738	. . . 4 |- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> ( 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 ) ) )
12	11	rneqd 5353	. . 3 |- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> 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 ) ) )
13	12	infeq1d 8383	. 2 |- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> 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 ) , < ) )
14		simp3 1063	. . . 4 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> A C_ U. Q )
15		fdm 6051	. . . . . 6 |- ( R : Q --> ( 0 [,] +oo ) -> dom R = Q )
16	15	3ad2ant2 1083	. . . . 5 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> dom R = Q )
17	16	unieqd 4446	. . . 4 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> U. dom R = U. Q )
18	14, 17	sseqtr4d 3642	. . 3 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> A C_ U. dom R )
19		elex 3212	. . . . . 6 |- ( Q e. V -> Q e. _V )
20		uniexb 6973	. . . . . . 7 |- ( Q e. _V <-> U. Q e. _V )
21	20	biimpi 206	. . . . . 6 |- ( Q e. _V -> U. Q e. _V )
22	2, 19, 21	3syl 18	. . . . 5 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> U. Q e. _V )
23		ssexg 4804	. . . . 5 |- ( ( A C_ U. Q /\ U. Q e. _V ) -> A e. _V )
24	14, 22, 23	syl2anc 693	. . . 4 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> A e. _V )
25		elpwg 4166	. . . 4 |- ( A e. _V -> ( A e. ~P U. dom R <-> A C_ U. dom R ) )
26	24, 25	syl 17	. . 3 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> ( A e. ~P U. dom R <-> A C_ U. dom R ) )
27	18, 26	mpbird 247	. 2 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> A e. ~P U. dom R )
28		xrltso 11974	. . . 4 |- < Or RR*
29		iccssxr 12256	. . . . 5 |- ( 0 [,] +oo ) C_ RR*
30		soss 5053	. . . . 5 |- ( ( 0 [,] +oo ) C_ RR* -> ( < Or RR* -> < Or ( 0 [,] +oo ) ) )
31	29, 30	ax-mp 5	. . . 4 |- ( < Or RR* -> < Or ( 0 [,] +oo ) )
32	28, 31	mp1i 13	. . 3 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> < Or ( 0 [,] +oo ) )
33	32	infexd 8389	. 2 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> 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 )
34	6, 13, 27, 33	fvmptd 6288	1 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> ( (toOMeas ` R ) ` A ) = 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   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   Or wor 5034   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953  infcinf 8347   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < 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  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-po 5035  df-so 5036  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-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-icc 12182  df-esum 30090  df-oms 30354

This theorem is referenced by:  omsf  30358  oms0  30359  omsmon  30360  omssubaddlem  30361  omssubadd  30362