Theorem caragenel 46524

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 46522	. . . . 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 2782	. . 3 ⊢ (𝜑 → 𝑆 = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
6	5	eleq2d 2820	. 2 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ 𝐸 ∈ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}))
7		ineq2 4189	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (𝑎 ∩ 𝑒) = (𝑎 ∩ 𝐸))
8	7	fveq2d 6880	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (𝑂‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝐸)))
9		difeq2 4095	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (𝑎 ∖ 𝑒) = (𝑎 ∖ 𝐸))
10	9	fveq2d 6880	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (𝑂‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝐸)))
11	8, 10	oveq12d 7423	. . . . . 6 ⊢ (𝑒 = 𝐸 → ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))))
12	11	eqeq1d 2737	. . . . 5 ⊢ (𝑒 = 𝐸 → (((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
13	12	ralbidv 3163	. . . 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 3671	. . 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 279	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  {crab 3415   ∖ cdif 3923   ∩ cin 3925  𝒫 cpw 4575  ∪ cuni 4883  dom cdm 5654  ‘cfv 6531  (class class class)co 7405   +_𝑒 cxad 13126  OutMeascome 46518  CaraGenccaragen 46520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-caragen 46521

This theorem is referenced by:  caragensplit  46529  caragenelss  46530  carageneld  46531  caragendifcl  46543  isvonmbl  46667