Theorem fmlafv 35743

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 35708	. . 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 6871	. . . 4 ⊢ (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
4	3	dmeqd 5886	. . 3 ⊢ (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
5	4	adantl 486	. 2 ⊢ ((𝑁 ∈ suc ω ∧ 𝑛 = 𝑁) → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
6		id 23	. 2 ⊢ (𝑁 ∈ suc ω → 𝑁 ∈ suc ω)
7		fvex 6884	. . . 4 ⊢ ((∅ Sat ∅)‘𝑁) ∈ V
8	7	dmex 7894	. . 3 ⊢ dom ((∅ Sat ∅)‘𝑁) ∈ V
9	8	a1i 11	. 2 ⊢ (𝑁 ∈ suc ω → dom ((∅ Sat ∅)‘𝑁) ∈ V)
10	2, 5, 6, 9	fvmptd 6987	1 ⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ∅c0 4288   ↦ cmpt 5186  dom cdm 5652  suc csuc 6352  ‘cfv 6525  (class class class)co 7400  ωcom 7850   Sat csat 35699  Fmlacfmla 35700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-fmla 35708

This theorem is referenced by:  fmla  35744  fmla0  35745  fmlasuc0  35747  satfdmfmla  35763