Theorem caragenel 42771

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 42769	. . . . 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 17	. . . 4 ⊢ (𝜑 → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
5	1, 4	syl5eq 2868	. . 3 ⊢ (𝜑 → 𝑆 = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
6	5	eleq2d 2898	. 2 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ 𝐸 ∈ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}))
7		ineq2 4182	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (𝑎 ∩ 𝑒) = (𝑎 ∩ 𝐸))
8	7	fveq2d 6668	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (𝑂‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝐸)))
9		difeq2 4092	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (𝑎 ∖ 𝑒) = (𝑎 ∖ 𝐸))
10	9	fveq2d 6668	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (𝑂‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝐸)))
11	8, 10	oveq12d 7168	. . . . . 6 ⊢ (𝑒 = 𝐸 → ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))))
12	11	eqeq1d 2823	. . . . 5 ⊢ (𝑒 = 𝐸 → (((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
13	12	ralbidv 3197	. . . 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 3679	. . 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 281	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 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142   ∖ cdif 3932   ∩ cin 3934  𝒫 cpw 4538  ∪ cuni 4831  dom cdm 5549  ‘cfv 6349  (class class class)co 7150   +_𝑒 cxad 12499  OutMeascome 42765  CaraGenccaragen 42767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153  df-caragen 42768

This theorem is referenced by:  caragensplit  42776  caragenelss  42777  carageneld  42778  caragendifcl  42790  isvonmbl  42914