Theorem carageneld 40716

Description: Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
carageneld.o	|- ( ph -> O e. OutMeas )
carageneld.x	|- X = U. dom O
carageneld.s	|- S = (CaraGen ` O )
carageneld.e	|- ( ph -> E e. ~P X )
carageneld.a	|- ( ( ph /\ a e. ~P X ) -> ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) )

Assertion

Ref	Expression
carageneld	|- ( ph -> E e. S )

Distinct variable groups:   E, a   O, a   ph, a

Allowed substitution hints:   S( a)   X( a)

Proof of Theorem carageneld

Step	Hyp	Ref	Expression
1		carageneld.e	. . . 4 |- ( ph -> E e. ~P X )
2		carageneld.x	. . . . 5 |- X = U. dom O
3	2	pweqi 4162	. . . 4 |- ~P X = ~P U. dom O
4	1, 3	syl6eleq 2711	. . 3 |- ( ph -> E e. ~P U. dom O )
5		simpl 473	. . . . 5 |- ( ( ph /\ a e. ~P U. dom O ) -> ph )
6	3	eleq2i 2693	. . . . . . . 8 |- ( a e. ~P X <-> a e. ~P U. dom O )
7	6	bicomi 214	. . . . . . 7 |- ( a e. ~P U. dom O <-> a e. ~P X )
8	7	biimpi 206	. . . . . 6 |- ( a e. ~P U. dom O -> a e. ~P X )
9	8	adantl 482	. . . . 5 |- ( ( ph /\ a e. ~P U. dom O ) -> a e. ~P X )
10		carageneld.a	. . . . 5 |- ( ( ph /\ a e. ~P X ) -> ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) )
11	5, 9, 10	syl2anc 693	. . . 4 |- ( ( ph /\ a e. ~P U. dom O ) -> ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) )
12	11	ralrimiva 2966	. . 3 |- ( ph -> A. a e. ~P U. dom O ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) )
13	4, 12	jca 554	. 2 |- ( ph -> ( E e. ~P U. dom O /\ A. a e. ~P U. dom O ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) ) )
14		carageneld.o	. . 3 |- ( ph -> O e. OutMeas )
15		carageneld.s	. . 3 |- S = (CaraGen ` O )
16	14, 15	caragenel 40709	. 2 |- ( ph -> ( E e. S <-> ( E e. ~P U. dom O /\ A. a e. ~P U. dom O ( ( O ` ( a i^i E ) ) +e ( O ` ( a \ E ) ) ) = ( O ` a ) ) ) )
17	13, 16	mpbird 247	1 |- ( ph -> E e. S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   i^i cin 3573   ~Pcpw 4158   U.cuni 4436   dom cdm 5114   ` cfv 5888  (class class class)co 6650   +ecxad 11944  OutMeascome 40703  CaraGenccaragen 40705

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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-caragen 40706

This theorem is referenced by:  caragen0  40720  caragenunidm  40722  caragenuncl  40727  caragendifcl  40728  carageniuncl  40737  caragenel2d  40746