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Theorem fmlafv 35350
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 35315 . . 3 Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
21a1i 11 . 2 (𝑁 ∈ suc ω → Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)))
3 fveq2 6922 . . . 4 (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
43dmeqd 5930 . . 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 6935 . . . 4 ((∅ Sat ∅)‘𝑁) ∈ V
87dmex 7951 . . 3 dom ((∅ Sat ∅)‘𝑁) ∈ V
98a1i 11 . 2 (𝑁 ∈ suc ω → dom ((∅ Sat ∅)‘𝑁) ∈ V)
102, 5, 6, 9fvmptd 7038 1 (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  Vcvv 3488  c0 4352  cmpt 5249  dom cdm 5700  suc csuc 6399  cfv 6575  (class class class)co 7450  ωcom 7905   Sat csat 35306  Fmlacfmla 35307
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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7772
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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6527  df-fun 6577  df-fv 6583  df-fmla 35315
This theorem is referenced by:  fmla  35351  fmla0  35352  fmlasuc0  35354  satfdmfmla  35370
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