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Theorem satefv 33811
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 33741 . . 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 5663 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚 × 𝑚) = (𝑀 × 𝑀))
54ineq2d 4170 . . . . . . 7 (𝑚 = 𝑀 → ( E ∩ (𝑚 × 𝑚)) = ( E ∩ (𝑀 × 𝑀)))
63, 5oveq12d 7369 . . . . . 6 (𝑚 = 𝑀 → (𝑚 Sat ( E ∩ (𝑚 × 𝑚))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀))))
76fveq1d 6841 . . . . 5 (𝑚 = 𝑀 → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
87adantr 481 . . . 4 ((𝑚 = 𝑀𝑢 = 𝑈) → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
9 simpr 485 . . . 4 ((𝑚 = 𝑀𝑢 = 𝑈) → 𝑢 = 𝑈)
108, 9fveq12d 6846 . . 3 ((𝑚 = 𝑀𝑢 = 𝑈) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
1110adantl 482 . 2 (((𝑀𝑉𝑈𝑊) ∧ (𝑚 = 𝑀𝑢 = 𝑈)) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
12 elex 3461 . . 3 (𝑀𝑉𝑀 ∈ V)
1312adantr 481 . 2 ((𝑀𝑉𝑈𝑊) → 𝑀 ∈ V)
14 elex 3461 . . 3 (𝑈𝑊𝑈 ∈ V)
1514adantl 482 . 2 ((𝑀𝑉𝑈𝑊) → 𝑈 ∈ V)
16 fvexd 6854 . 2 ((𝑀𝑉𝑈𝑊) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈) ∈ V)
172, 11, 13, 15, 16ovmpod 7501 1 ((𝑀𝑉𝑈𝑊) → (𝑀 Sat 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3443  cin 3907   E cep 5534   × cxp 5629  cfv 6493  (class class class)co 7351  cmpo 7353  ωcom 7794   Sat csat 33733   Sat csate 33735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-sate 33741
This theorem is referenced by:  sate0  33812  satef  33813  satefvfmla0  33815  satefvfmla1  33822
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