Theorem caragenel 47161

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

Hypotheses

Ref	Expression
caragenel.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
caragenel.s	⊢ 𝑆 = (CaraGen‘𝑂)

Assertion

Ref	Expression
caragenel	⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))))

Distinct variable groups:   𝐸,𝑎   𝑂,𝑎

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

Proof of Theorem caragenel

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caragenel.s	. . . 4 ⊢ 𝑆 = (CaraGen‘𝑂)
2		caragenel.o	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
3		caragenval 47159	. . . . 5 ⊢ (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
4	2, 3	syl 18	. . . 4 ⊢ (𝜑 → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
5	1, 4	eqtrid 2817	. . 3 ⊢ (𝜑 → 𝑆 = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
6	5	eleq2d 2856	. 2 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ 𝐸 ∈ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}))
7		ineq2 4175	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (𝑎 ∩ 𝑒) = (𝑎 ∩ 𝐸))
8	7	fveq2d 6889	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (𝑂‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝐸)))
9		difeq2 4083	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (𝑎 ∖ 𝑒) = (𝑎 ∖ 𝐸))
10	9	fveq2d 6889	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (𝑂‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝐸)))
11	8, 10	oveq12d 7432	. . . . . 6 ⊢ (𝑒 = 𝐸 → ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))))
12	11	eqeq1d 2772	. . . . 5 ⊢ (𝑒 = 𝐸 → (((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
13	12	ralbidv 3195	. . . 4 ⊢ (𝑒 = 𝐸 → (∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎) ↔ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
14	13	elrab 3658	. . 3 ⊢ (𝐸 ∈ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
15	14	a1i 11	. 2 ⊢ (𝜑 → (𝐸 ∈ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))))
16	6, 15	bitrd 282	1 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2150  ∀wral 3086  {crab 3423   ∖ cdif 3910   ∩ cin 3912  𝒫 cpw 4567  ∪ cuni 4877  dom cdm 5665  ‘cfv 6540  (class class class)co 7414   +_𝑒 cxad 13138  OutMeascome 47155  CaraGenccaragen 47157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7417  df-caragen 47158

This theorem is referenced by:  caragensplit  47166  caragenelss  47167  carageneld  47168  caragendifcl  47180  isvonmbl  47304