Theorem carsgval 34935

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 34934	. 2 ⊢ toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +e (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)})
2		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
3	2	dmeqd 5886	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
4		carsgval.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
5	4	fdmd 6720	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 = 𝒫 𝑂)
6	5	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → dom 𝑀 = 𝒫 𝑂)
7	3, 6	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → dom 𝑚 = 𝒫 𝑂)
8	7	unieqd 4879	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ 𝒫 𝑂)
9		unipw 5417	. . . . 5 ⊢ ∪ 𝒫 𝑂 = 𝑂
10	8, 9	eqtrdi 2811	. . . 4 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = 𝑂)
11	10	pweqd 4573	. . 3 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → 𝒫 ∪ dom 𝑚 = 𝒫 𝑂)
12		fveq1 6884	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑒 ∩ 𝑎)) = (𝑀‘(𝑒 ∩ 𝑎)))
13		fveq1 6884	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑒 ∖ 𝑎)) = (𝑀‘(𝑒 ∖ 𝑎)))
14	12, 13	oveq12d 7438	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝑚‘(𝑒 ∩ 𝑎)) +e (𝑚‘(𝑒 ∖ 𝑎))) = ((𝑀‘(𝑒 ∩ 𝑎)) +e (𝑀‘(𝑒 ∖ 𝑎))))
15		fveq1 6884	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚‘𝑒) = (𝑀‘𝑒))
16	14, 15	eqeq12d 2776	. . . . 5 ⊢ (𝑚 = 𝑀 → (((𝑚‘(𝑒 ∩ 𝑎)) +e (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒) ↔ ((𝑀‘(𝑒 ∩ 𝑎)) +e (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)))
17	16	adantl 487	. . . 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 3334	. . 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 3429	. 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 5340	. . 3 ⊢ (𝜑 → 𝒫 𝑂 ∈ V)
22	4, 21	fexd 7233	. 2 ⊢ (𝜑 → 𝑀 ∈ V)
23		pwexg 5339	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V)
24		rabexg 5298	. . 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 19	. 2 ⊢ (𝜑 → {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +e (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)} ∈ V)
26	1, 19, 22, 25	fvmptd2 7002	1 ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +e (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ∖ cdif 4045   ∩ cin 4089  𝒫 cpw 4556  ∪ cuni 4866  dom cdm 5650  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  0cc0 11200  +∞cpnf 11340   +_e cxad 13239  [,]cicc 13479  toCaraSigaccarsg 34933

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-carsg 34934

This theorem is used by:  carsgcl  34936  elcarsg  34937