Theorem carsgval 30906

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 30905	. . 3 ⊢ toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)})
2	1	a1i 11	. 2 ⊢ (𝜑 → toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)}))
3		simpr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
4	3	dmeqd 5562	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
5		carsgval.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
6	5	fdmd 6291	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 = 𝒫 𝑂)
7	6	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → dom 𝑀 = 𝒫 𝑂)
8	4, 7	eqtrd 2861	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → dom 𝑚 = 𝒫 𝑂)
9	8	unieqd 4670	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = ∪ 𝒫 𝑂)
10		unipw 5141	. . . . 5 ⊢ ∪ 𝒫 𝑂 = 𝑂
11	9, 10	syl6eq 2877	. . . 4 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → ∪ dom 𝑚 = 𝑂)
12	11	pweqd 4385	. . 3 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → 𝒫 ∪ dom 𝑚 = 𝒫 𝑂)
13		fveq1 6436	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑒 ∩ 𝑎)) = (𝑀‘(𝑒 ∩ 𝑎)))
14		fveq1 6436	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚‘(𝑒 ∖ 𝑎)) = (𝑀‘(𝑒 ∖ 𝑎)))
15	13, 14	oveq12d 6928	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = ((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))))
16		fveq1 6436	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚‘𝑒) = (𝑀‘𝑒))
17	15, 16	eqeq12d 2840	. . . . 5 ⊢ (𝑚 = 𝑀 → (((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒) ↔ ((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)))
18	17	adantl 475	. . . 4 ⊢ ((𝜑 ∧ 𝑚 = 𝑀) → (((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒) ↔ ((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)))
19	12, 18	raleqbidv 3364	. . 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)))
20	12, 19	rabeqbidv 3408	. 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)})
21		carsgval.1	. . . 4 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
22	21	pwexd 5081	. . 3 ⊢ (𝜑 → 𝒫 𝑂 ∈ V)
23		fex 6750	. . 3 ⊢ ((𝑀:𝒫 𝑂⟶(0[,]+∞) ∧ 𝒫 𝑂 ∈ V) → 𝑀 ∈ V)
24	5, 22, 23	syl2anc 579	. 2 ⊢ (𝜑 → 𝑀 ∈ V)
25		pwexg 5080	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V)
26		rabexg 5038	. . 3 ⊢ (𝒫 𝑂 ∈ V → {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)} ∈ V)
27	21, 25, 26	3syl 18	. 2 ⊢ (𝜑 → {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)} ∈ V)
28	2, 20, 24, 27	fvmptd 6539	1 ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)})

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∀wral 3117  {crab 3121  Vcvv 3414   ∖ cdif 3795   ∩ cin 3797  𝒫 cpw 4380  ∪ cuni 4660   ↦ cmpt 4954  dom cdm 5346  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  0cc0 10259  +∞cpnf 10395   +_𝑒 cxad 12237  [,]cicc 12473  toCaraSigaccarsg 30904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-carsg 30905

This theorem is referenced by:  carsgcl  30907  elcarsg  30908