Theorem carsgval 34335

Description: Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.)

Hypotheses

Ref	Expression
carsgval.1	⊢ (𝜑 → 𝑂 ∈ 𝑉)
carsgval.2	⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))

Assertion

Ref	Expression
carsgval	⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)})

Distinct variable groups:   𝑀,𝑎,𝑒   𝑂,𝑎,𝑒   𝜑,𝑎,𝑒

Allowed substitution hints:   𝑉(𝑒,𝑎)

Proof of Theorem carsgval

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-carsg 34334	. 2 ⊢ toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)})
2		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
3	2	dmeqd 5885	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
4		carsgval.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
5	4	fdmd 6716	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 = 𝒫 𝑂)
6	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → dom 𝑀 = 𝒫 𝑂)
7	3, 6	eqtrd 2770	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → dom 𝑚 = 𝒫 𝑂)
8	7	unieqd 4896	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ 𝒫 𝑂)
9		unipw 5425	. . . . 5 ⊢ ∪ 𝒫 𝑂 = 𝑂
10	8, 9	eqtrdi 2786	. . . 4 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = 𝑂)
11	10	pweqd 4592	. . 3 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → 𝒫 ∪ dom 𝑚 = 𝒫 𝑂)
12		fveq1 6875	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑒 ∩ 𝑎)) = (𝑀‘(𝑒 ∩ 𝑎)))
13		fveq1 6875	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑒 ∖ 𝑎)) = (𝑀‘(𝑒 ∖ 𝑎)))
14	12, 13	oveq12d 7423	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = ((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))))
15		fveq1 6875	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚‘𝑒) = (𝑀‘𝑒))
16	14, 15	eqeq12d 2751	. . . . 5 ⊢ (𝑚 = 𝑀 → (((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒) ↔ ((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)))
17	16	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → (((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒) ↔ ((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)))
18	11, 17	raleqbidv 3325	. . 3 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → (∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒) ↔ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)))
19	11, 18	rabeqbidv 3434	. 2 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)} = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)})
20		carsgval.1	. . . 4 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
21	20	pwexd 5349	. . 3 ⊢ (𝜑 → 𝒫 𝑂 ∈ V)
22	4, 21	fexd 7219	. 2 ⊢ (𝜑 → 𝑀 ∈ V)
23		pwexg 5348	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V)
24		rabexg 5307	. . 3 ⊢ (𝒫 𝑂 ∈ V → {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)} ∈ V)
25	20, 23, 24	3syl 18	. 2 ⊢ (𝜑 → {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)} ∈ V)
26	1, 19, 22, 25	fvmptd2 6994	1 ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  {crab 3415  Vcvv 3459   ∖ cdif 3923   ∩ cin 3925  𝒫 cpw 4575  ∪ cuni 4883  dom cdm 5654  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  0cc0 11129  +∞cpnf 11266   +_𝑒 cxad 13126  [,]cicc 13365  toCaraSigaccarsg 34333

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-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402

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-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  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-iun 4969  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-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-carsg 34334

This theorem is referenced by:  carsgcl  34336  elcarsg  34337