Theorem fmlafv 34669

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 34634	. . 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 6890	. . . 4 ⊢ (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
4	3	dmeqd 5904	. . 3 ⊢ (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
5	4	adantl 480	. 2 ⊢ ((𝑁 ∈ suc ω ∧ 𝑛 = 𝑁) → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
6		id 22	. 2 ⊢ (𝑁 ∈ suc ω → 𝑁 ∈ suc ω)
7		fvex 6903	. . . 4 ⊢ ((∅ Sat ∅)‘𝑁) ∈ V
8	7	dmex 7904	. . 3 ⊢ dom ((∅ Sat ∅)‘𝑁) ∈ V
9	8	a1i 11	. 2 ⊢ (𝑁 ∈ suc ω → dom ((∅ Sat ∅)‘𝑁) ∈ V)
10	2, 5, 6, 9	fvmptd 7004	1 ⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  Vcvv 3472  ∅c0 4321   ↦ cmpt 5230  dom cdm 5675  suc csuc 6365  ‘cfv 6542  (class class class)co 7411  ωcom 7857   Sat csat 34625  Fmlacfmla 34626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fv 6550  df-fmla 34634

This theorem is referenced by:  fmla  34670  fmla0  34671  fmlasuc0  34673  satfdmfmla  34689