Theorem carsgval 30365

Description: Value of the Caratheodory sigma-Algebra construction function. (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
carsgval	|- ( ph -> (toCaraSiga ` M ) = { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } )

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

Allowed substitution hints:   V( e, a)

Proof of Theorem carsgval

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-carsg 30364	. . 3 |- toCaraSiga = ( m e. _V |-> { a e. ~P U. dom m | A. e e. ~P U. dom m ( ( m ` ( e i^i a ) ) +e ( m ` ( e \ a ) ) ) = ( m ` e ) } )
2	1	a1i 11	. 2 |- ( ph -> toCaraSiga = ( m e. _V |-> { a e. ~P U. dom m | A. e e. ~P U. dom m ( ( m ` ( e i^i a ) ) +e ( m ` ( e \ a ) ) ) = ( m ` e ) } ) )
3		simpr 477	. . . . . . . 8 |- ( ( ph /\ m = M ) -> m = M )
4	3	dmeqd 5326	. . . . . . 7 |- ( ( ph /\ m = M ) -> dom m = dom M )
5		carsgval.2	. . . . . . . . 9 |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )
6		fdm 6051	. . . . . . . . 9 |- ( M : ~P O --> ( 0 [,] +oo ) -> dom M = ~P O )
7	5, 6	syl 17	. . . . . . . 8 |- ( ph -> dom M = ~P O )
8	7	adantr 481	. . . . . . 7 |- ( ( ph /\ m = M ) -> dom M = ~P O )
9	4, 8	eqtrd 2656	. . . . . 6 |- ( ( ph /\ m = M ) -> dom m = ~P O )
10	9	unieqd 4446	. . . . 5 |- ( ( ph /\ m = M ) -> U. dom m = U. ~P O )
11		unipw 4918	. . . . 5 |- U. ~P O = O
12	10, 11	syl6eq 2672	. . . 4 |- ( ( ph /\ m = M ) -> U. dom m = O )
13	12	pweqd 4163	. . 3 |- ( ( ph /\ m = M ) -> ~P U. dom m = ~P O )
14		fveq1 6190	. . . . . . 7 |- ( m = M -> ( m ` ( e i^i a ) ) = ( M ` ( e i^i a ) ) )
15		fveq1 6190	. . . . . . 7 |- ( m = M -> ( m ` ( e \ a ) ) = ( M ` ( e \ a ) ) )
16	14, 15	oveq12d 6668	. . . . . 6 |- ( m = M -> ( ( m ` ( e i^i a ) ) +e ( m ` ( e \ a ) ) ) = ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) )
17		fveq1 6190	. . . . . 6 |- ( m = M -> ( m ` e ) = ( M ` e ) )
18	16, 17	eqeq12d 2637	. . . . 5 |- ( m = M -> ( ( ( m ` ( e i^i a ) ) +e ( m ` ( e \ a ) ) ) = ( m ` e ) <-> ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) ) )
19	18	adantl 482	. . . 4 |- ( ( ph /\ m = M ) -> ( ( ( m ` ( e i^i a ) ) +e ( m ` ( e \ a ) ) ) = ( m ` e ) <-> ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) ) )
20	13, 19	raleqbidv 3152	. . 3 |- ( ( ph /\ m = M ) -> ( A. e e. ~P U. dom m ( ( 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 ) ) )
21	13, 20	rabeqbidv 3195	. 2 |- ( ( ph /\ m = M ) -> { a e. ~P U. dom m | A. e e. ~P U. dom m ( ( 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 ) } )
22		carsgval.1	. . . 4 |- ( ph -> O e. V )
23		pwexg 4850	. . . 4 |- ( O e. V -> ~P O e. _V )
24	22, 23	syl 17	. . 3 |- ( ph -> ~P O e. _V )
25		fex 6490	. . 3 |- ( ( M : ~P O --> ( 0 [,] +oo ) /\ ~P O e. _V ) -> M e. _V )
26	5, 24, 25	syl2anc 693	. 2 |- ( ph -> M e. _V )
27		rabexg 4812	. . 3 |- ( ~P O e. _V -> { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } e. _V )
28	22, 23, 27	3syl 18	. 2 |- ( ph -> { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } e. _V )
29	2, 21, 26, 28	fvmptd 6288	1 |- ( ph -> (toCaraSiga ` M ) = { a e. ~P 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   ~Pcpw 4158   U.cuni 4436   |-> cmpt 4729   dom cdm 5114   -->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:  carsgcl  30366  elcarsg  30367