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Theorem fmlafv 35340
Description: The valid Godel formulas of height 𝑁 is the domain of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 15-Sep-2023.)
Assertion
Ref Expression
fmlafv (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))

Proof of Theorem fmlafv
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 df-fmla 35305 . . 3 Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
21a1i 11 . 2 (𝑁 ∈ suc ω → Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)))
3 fveq2 6919 . . . 4 (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
43dmeqd 5929 . . 3 (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
54adantl 481 . 2 ((𝑁 ∈ suc ω ∧ 𝑛 = 𝑁) → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
6 id 22 . 2 (𝑁 ∈ suc ω → 𝑁 ∈ suc ω)
7 fvex 6932 . . . 4 ((∅ Sat ∅)‘𝑁) ∈ V
87dmex 7945 . . 3 dom ((∅ Sat ∅)‘𝑁) ∈ V
98a1i 11 . 2 (𝑁 ∈ suc ω → dom ((∅ Sat ∅)‘𝑁) ∈ V)
102, 5, 6, 9fvmptd 7034 1 (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2103  Vcvv 3482  c0 4347  cmpt 5252  dom cdm 5699  suc csuc 6396  cfv 6572  (class class class)co 7445  ωcom 7899   Sat csat 35296  Fmlacfmla 35297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-iota 6524  df-fun 6574  df-fv 6580  df-fmla 35305
This theorem is referenced by:  fmla  35341  fmla0  35342  fmlasuc0  35344  satfdmfmla  35360
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