Theorem fmlafv 35967

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.

Step	Hyp	Ref	Expression
1		df-fmla 35932	. . 3 ⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
2	1	a1i 11	. 2 ⊢ (𝑁 ∈ suc ω → Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)))
3		fveq2 6882	. . . 4 ⊢ (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
4	3	dmeqd 5893	. . 3 ⊢ (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
5	4	adantl 487	. 2 ⊢ ((𝑁 ∈ suc ω ∧ 𝑛 = 𝑁) → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
6		id 23	. 2 ⊢ (𝑁 ∈ suc ω → 𝑁 ∈ suc ω)
7		fvex 6895	. . . 4 ⊢ ((∅ Sat ∅)‘𝑁) ∈ V
8	7	dmex 7910	. . 3 ⊢ dom ((∅ Sat ∅)‘𝑁) ∈ V
9	8	a1i 11	. 2 ⊢ (𝑁 ∈ suc ω → dom ((∅ Sat ∅)‘𝑁) ∈ V)
10	2, 5, 6, 9	fvmptd 6998	1 ⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ∅c0 4282   ↦ cmpt 5190  dom cdm 5659  suc csuc 6363  ‘cfv 6537  (class class class)co 7417  ωcom 7866   Sat csat 35923  Fmlacfmla 35924

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fv 6545  df-fmla 35932

This theorem is used by:  fmla  35968  fmla0  35969  fmlasuc0  35971  satfdmfmla  35987