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Theorem carsgval 30906
Description: Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.)
Hypotheses
Ref Expression
carsgval.1 (𝜑𝑂𝑉)
carsgval.2 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
Assertion
Ref Expression
carsgval (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)})
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.
StepHypRef Expression
1 df-carsg 30905 . . 3 toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 dom 𝑚((𝑚‘(𝑒𝑎)) +𝑒 (𝑚‘(𝑒𝑎))) = (𝑚𝑒)})
21a1i 11 . 2 (𝜑 → toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 dom 𝑚((𝑚‘(𝑒𝑎)) +𝑒 (𝑚‘(𝑒𝑎))) = (𝑚𝑒)}))
3 simpr 479 . . . . . . . 8 ((𝜑𝑚 = 𝑀) → 𝑚 = 𝑀)
43dmeqd 5562 . . . . . . 7 ((𝜑𝑚 = 𝑀) → dom 𝑚 = dom 𝑀)
5 carsgval.2 . . . . . . . . 9 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
65fdmd 6291 . . . . . . . 8 (𝜑 → dom 𝑀 = 𝒫 𝑂)
76adantr 474 . . . . . . 7 ((𝜑𝑚 = 𝑀) → dom 𝑀 = 𝒫 𝑂)
84, 7eqtrd 2861 . . . . . 6 ((𝜑𝑚 = 𝑀) → dom 𝑚 = 𝒫 𝑂)
98unieqd 4670 . . . . 5 ((𝜑𝑚 = 𝑀) → dom 𝑚 = 𝒫 𝑂)
10 unipw 5141 . . . . 5 𝒫 𝑂 = 𝑂
119, 10syl6eq 2877 . . . 4 ((𝜑𝑚 = 𝑀) → dom 𝑚 = 𝑂)
1211pweqd 4385 . . 3 ((𝜑𝑚 = 𝑀) → 𝒫 dom 𝑚 = 𝒫 𝑂)
13 fveq1 6436 . . . . . . 7 (𝑚 = 𝑀 → (𝑚‘(𝑒𝑎)) = (𝑀‘(𝑒𝑎)))
14 fveq1 6436 . . . . . . 7 (𝑚 = 𝑀 → (𝑚‘(𝑒𝑎)) = (𝑀‘(𝑒𝑎)))
1513, 14oveq12d 6928 . . . . . 6 (𝑚 = 𝑀 → ((𝑚‘(𝑒𝑎)) +𝑒 (𝑚‘(𝑒𝑎))) = ((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))))
16 fveq1 6436 . . . . . 6 (𝑚 = 𝑀 → (𝑚𝑒) = (𝑀𝑒))
1715, 16eqeq12d 2840 . . . . 5 (𝑚 = 𝑀 → (((𝑚‘(𝑒𝑎)) +𝑒 (𝑚‘(𝑒𝑎))) = (𝑚𝑒) ↔ ((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)))
1817adantl 475 . . . 4 ((𝜑𝑚 = 𝑀) → (((𝑚‘(𝑒𝑎)) +𝑒 (𝑚‘(𝑒𝑎))) = (𝑚𝑒) ↔ ((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)))
1912, 18raleqbidv 3364 . . 3 ((𝜑𝑚 = 𝑀) → (∀𝑒 ∈ 𝒫 dom 𝑚((𝑚‘(𝑒𝑎)) +𝑒 (𝑚‘(𝑒𝑎))) = (𝑚𝑒) ↔ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)))
2012, 19rabeqbidv 3408 . 2 ((𝜑𝑚 = 𝑀) → {𝑎 ∈ 𝒫 dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 dom 𝑚((𝑚‘(𝑒𝑎)) +𝑒 (𝑚‘(𝑒𝑎))) = (𝑚𝑒)} = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)})
21 carsgval.1 . . . 4 (𝜑𝑂𝑉)
2221pwexd 5081 . . 3 (𝜑 → 𝒫 𝑂 ∈ V)
23 fex 6750 . . 3 ((𝑀:𝒫 𝑂⟶(0[,]+∞) ∧ 𝒫 𝑂 ∈ V) → 𝑀 ∈ V)
245, 22, 23syl2anc 579 . 2 (𝜑𝑀 ∈ V)
25 pwexg 5080 . . 3 (𝑂𝑉 → 𝒫 𝑂 ∈ V)
26 rabexg 5038 . . 3 (𝒫 𝑂 ∈ V → {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)} ∈ V)
2721, 25, 263syl 18 . 2 (𝜑 → {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)} ∈ V)
282, 20, 24, 27fvmptd 6539 1 (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)})
Colors of variables: wff setvar class
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
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