Theorem satfdmfmla 35598

Description: The domain of the satisfaction predicate as function over wff codes in any model 𝑀 and any binary relation 𝐸 on 𝑀 for a natural number 𝑁 is the set of valid Godel formulas of height 𝑁. (Contributed by AV, 13-Oct-2023.)

Assertion

Ref	Expression
satfdmfmla	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))

Proof of Theorem satfdmfmla

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5242	. . . . . . 7 ⊢ ∅ ∈ V
2	1, 1	pm3.2i 470	. . . . . 6 ⊢ (∅ ∈ V ∧ ∅ ∈ V)
3	2	jctr 524	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
4	3	3adant3 1133	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
5		satfdm 35567	. . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛))
6	4, 5	syl 17	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛))
7		fveq2 6834	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑀 Sat 𝐸)‘𝑛) = ((𝑀 Sat 𝐸)‘𝑁))
8	7	dmeqd 5854	. . . . . 6 ⊢ (𝑛 = 𝑁 → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑀 Sat 𝐸)‘𝑁))
9		fveq2 6834	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
10	9	dmeqd 5854	. . . . . 6 ⊢ (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
11	8, 10	eqeq12d 2753	. . . . 5 ⊢ (𝑛 = 𝑁 → (dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) ↔ dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
12	11	rspcv 3561	. . . 4 ⊢ (𝑁 ∈ ω → (∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
13	12	3ad2ant3 1136	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
14	6, 13	mpd 15	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
15		elelsuc 6392	. . . 4 ⊢ (𝑁 ∈ ω → 𝑁 ∈ suc ω)
16	15	3ad2ant3 1136	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → 𝑁 ∈ suc ω)
17		fmlafv 35578	. . 3 ⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
18	16, 17	syl 17	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
19	14, 18	eqtr4d 2775	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430  ∅c0 4274  dom cdm 5624  suc csuc 6319  ‘cfv 6492  (class class class)co 7360  ωcom 7810   Sat csat 35534  Fmlacfmla 35535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-goel 35538  df-goal 35540  df-sat 35541  df-fmla 35543

This theorem is referenced by:  satffunlem1lem2  35601  satffunlem2lem2  35604  satff  35608  satefvfmla0  35616  satefvfmla1  35623