Theorem fmlafv 35574

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 35539	. . 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 6834	. . . 4 ⊢ (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
4	3	dmeqd 5854	. . 3 ⊢ (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
5	4	adantl 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 6847	. . . 4 ⊢ ((∅ Sat ∅)‘𝑁) ∈ V
8	7	dmex 7851	. . 3 ⊢ dom ((∅ Sat ∅)‘𝑁) ∈ V
9	8	a1i 11	. 2 ⊢ (𝑁 ∈ suc ω → dom ((∅ Sat ∅)‘𝑁) ∈ V)
10	2, 5, 6, 9	fvmptd 6948	1 ⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285   ↦ cmpt 5179  dom cdm 5624  suc csuc 6319  ‘cfv 6492  (class class class)co 7358  ωcom 7808   Sat csat 35530  Fmlacfmla 35531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-fmla 35539

This theorem is referenced by:  fmla  35575  fmla0  35576  fmlasuc0  35578  satfdmfmla  35594