Theorem elcarsg 30367

Description: Property of being a Catatheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.)

Hypotheses

Ref	Expression
carsgval.1	|- ( ph -> O e. V )
carsgval.2	|- ( ph -> M : ~P O --> ( 0 [,] +oo ) )

Assertion

Ref	Expression
elcarsg	|- ( ph -> ( A e. (toCaraSiga ` M ) <-> ( A C_ O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) ) )

Distinct variable groups:   e, M   e, O   ph, e   A, e

Allowed substitution hint:   V( e)

Proof of Theorem elcarsg

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		carsgval.1	. . . 4 |- ( ph -> O e. V )
2		carsgval.2	. . . 4 |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )
3	1, 2	carsgval 30365	. . 3 |- ( ph -> (toCaraSiga ` M ) = { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } )
4	3	eleq2d 2687	. 2 |- ( ph -> ( A e. (toCaraSiga ` M ) <-> A e. { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } ) )
5		ineq2 3808	. . . . . . . 8 |- ( a = A -> ( e i^i a ) = ( e i^i A ) )
6	5	fveq2d 6195	. . . . . . 7 |- ( a = A -> ( M ` ( e i^i a ) ) = ( M ` ( e i^i A ) ) )
7		difeq2 3722	. . . . . . . 8 |- ( a = A -> ( e \ a ) = ( e \ A ) )
8	7	fveq2d 6195	. . . . . . 7 |- ( a = A -> ( M ` ( e \ a ) ) = ( M ` ( e \ A ) ) )
9	6, 8	oveq12d 6668	. . . . . 6 |- ( a = A -> ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) )
10	9	eqeq1d 2624	. . . . 5 |- ( a = A -> ( ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) <-> ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) )
11	10	ralbidv 2986	. . . 4 |- ( a = A -> ( A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) <-> A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) )
12	11	elrab 3363	. . 3 |- ( A e. { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } <-> ( A e. ~P O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) )
13		elex 3212	. . . . . 6 |- ( A e. ~P O -> A e. _V )
14	13	a1i 11	. . . . 5 |- ( ph -> ( A e. ~P O -> A e. _V ) )
15		simpr 477	. . . . . . 7 |- ( ( ph /\ A C_ O ) -> A C_ O )
16	1	adantr 481	. . . . . . 7 |- ( ( ph /\ A C_ O ) -> O e. V )
17		ssexg 4804	. . . . . . 7 |- ( ( A C_ O /\ O e. V ) -> A e. _V )
18	15, 16, 17	syl2anc 693	. . . . . 6 |- ( ( ph /\ A C_ O ) -> A e. _V )
19	18	ex 450	. . . . 5 |- ( ph -> ( A C_ O -> A e. _V ) )
20		elpwg 4166	. . . . . 6 |- ( A e. _V -> ( A e. ~P O <-> A C_ O ) )
21	20	a1i 11	. . . . 5 |- ( ph -> ( A e. _V -> ( A e. ~P O <-> A C_ O ) ) )
22	14, 19, 21	pm5.21ndd 369	. . . 4 |- ( ph -> ( A e. ~P O <-> A C_ O ) )
23	22	anbi1d 741	. . 3 |- ( ph -> ( ( A e. ~P O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) <-> ( A C_ O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) ) )
24	12, 23	syl5bb 272	. 2 |- ( ph -> ( A e. { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } <-> ( A C_ O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) ) )
25	4, 24	bitrd 268	1 |- ( ph -> ( A e. (toCaraSiga ` M ) <-> ( A C_ O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   +ecxad 11944   [,]cicc 12178  toCaraSigaccarsg 30363

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  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-carsg 30364

This theorem is referenced by:  baselcarsg  30368  0elcarsg  30369  difelcarsg  30372  inelcarsg  30373  carsgclctunlem1  30379  carsgclctunlem2  30381  carsgclctun  30383