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Theorem satefv 33089
Description: The simplified satisfaction predicate as function over wff codes in the model 𝑀 at the code 𝑈. (Contributed by AV, 30-Oct-2023.)
Assertion
Ref Expression
satefv ((𝑀𝑉𝑈𝑊) → (𝑀 Sat 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))

Proof of Theorem satefv
Dummy variables 𝑚 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sate 33019 . . 3 Sat = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))
21a1i 11 . 2 ((𝑀𝑉𝑈𝑊) → Sat = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢)))
3 id 22 . . . . . . 7 (𝑚 = 𝑀𝑚 = 𝑀)
43sqxpeqd 5583 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚 × 𝑚) = (𝑀 × 𝑀))
54ineq2d 4127 . . . . . . 7 (𝑚 = 𝑀 → ( E ∩ (𝑚 × 𝑚)) = ( E ∩ (𝑀 × 𝑀)))
63, 5oveq12d 7231 . . . . . 6 (𝑚 = 𝑀 → (𝑚 Sat ( E ∩ (𝑚 × 𝑚))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀))))
76fveq1d 6719 . . . . 5 (𝑚 = 𝑀 → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
87adantr 484 . . . 4 ((𝑚 = 𝑀𝑢 = 𝑈) → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
9 simpr 488 . . . 4 ((𝑚 = 𝑀𝑢 = 𝑈) → 𝑢 = 𝑈)
108, 9fveq12d 6724 . . 3 ((𝑚 = 𝑀𝑢 = 𝑈) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
1110adantl 485 . 2 (((𝑀𝑉𝑈𝑊) ∧ (𝑚 = 𝑀𝑢 = 𝑈)) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
12 elex 3426 . . 3 (𝑀𝑉𝑀 ∈ V)
1312adantr 484 . 2 ((𝑀𝑉𝑈𝑊) → 𝑀 ∈ V)
14 elex 3426 . . 3 (𝑈𝑊𝑈 ∈ V)
1514adantl 485 . 2 ((𝑀𝑉𝑈𝑊) → 𝑈 ∈ V)
16 fvexd 6732 . 2 ((𝑀𝑉𝑈𝑊) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈) ∈ V)
172, 11, 13, 15, 16ovmpod 7361 1 ((𝑀𝑉𝑈𝑊) → (𝑀 Sat 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  Vcvv 3408  cin 3865   E cep 5459   × cxp 5549  cfv 6380  (class class class)co 7213  cmpo 7215  ωcom 7644   Sat csat 33011   Sat csate 33013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-sate 33019
This theorem is referenced by:  sate0  33090  satef  33091  satefvfmla0  33093  satefvfmla1  33100
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