Theorem carageneld 47081

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

Hypotheses

Ref	Expression
carageneld.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
carageneld.x	⊢ 𝑋 = ∪ dom 𝑂
carageneld.s	⊢ 𝑆 = (CaraGen‘𝑂)
carageneld.e	⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋)
carageneld.a	⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))

Assertion

Ref	Expression
carageneld	⊢ (𝜑 → 𝐸 ∈ 𝑆)

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

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

Proof of Theorem carageneld

Step	Hyp	Ref	Expression
1		carageneld.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋)
2		carageneld.x	. . . . 5 ⊢ 𝑋 = ∪ dom 𝑂
3	2	pweqi 4573	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
4	1, 3	eleqtrdi 2874	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝒫 ∪ dom 𝑂)
5		simpl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝜑)
6	3	eleq2i 2856	. . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ∈ 𝒫 ∪ dom 𝑂)
7	6	bilanri 510	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ∈ 𝒫 𝑋)
8		carageneld.a	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
9	5, 7, 8	syl2anc 593	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
10	9	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
11	4, 10	jca 519	. 2 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
12		carageneld.o	. . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
13		carageneld.s	. . 3 ⊢ 𝑆 = (CaraGen‘𝑂)
14	12, 13	caragenel 47074	. 2 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))))
15	11, 14	mpbird 259	1 ⊢ (𝜑 → 𝐸 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ∖ cdif 3903   ∩ cin 3905  𝒫 cpw 4557  ∪ cuni 4867  dom cdm 5649  ‘cfv 6523  (class class class)co 7398   +_𝑒 cxad 13114  OutMeascome 47068  CaraGenccaragen 47070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401  df-caragen 47071

This theorem is referenced by:  caragen0  47085  caragenunidm  47087  caragenuncl  47092  caragendifcl  47093  carageniuncl  47102  caragenel2d  47111