Theorem caragenel 45946

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 45944	. . . . 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	eqtrid 2777	. . 3 ⊢ (𝜑 → 𝑆 = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
6	5	eleq2d 2811	. 2 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ 𝐸 ∈ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}))
7		ineq2 4200	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (𝑎 ∩ 𝑒) = (𝑎 ∩ 𝐸))
8	7	fveq2d 6896	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (𝑂‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝐸)))
9		difeq2 4108	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (𝑎 ∖ 𝑒) = (𝑎 ∖ 𝐸))
10	9	fveq2d 6896	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (𝑂‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝐸)))
11	8, 10	oveq12d 7434	. . . . . 6 ⊢ (𝑒 = 𝐸 → ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))))
12	11	eqeq1d 2727	. . . . 5 ⊢ (𝑒 = 𝐸 → (((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
13	12	ralbidv 3168	. . . 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 3674	. . 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 278	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 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3051  {crab 3419   ∖ cdif 3936   ∩ cin 3938  𝒫 cpw 4598  ∪ cuni 4903  dom cdm 5672  ‘cfv 6543  (class class class)co 7416   +_𝑒 cxad 13122  OutMeascome 45940  CaraGenccaragen 45942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7419  df-caragen 45943

This theorem is referenced by:  caragensplit  45951  caragenelss  45952  carageneld  45953  caragendifcl  45965  isvonmbl  46089