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Theorem satfdmfmla 33362
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 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))

Proof of Theorem satfdmfmla
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 0ex 5231 . . . . . . 7 ∅ ∈ V
21, 1pm3.2i 471 . . . . . 6 (∅ ∈ V ∧ ∅ ∈ V)
32jctr 525 . . . . 5 ((𝑀𝑉𝐸𝑊) → ((𝑀𝑉𝐸𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
433adant3 1131 . . . 4 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → ((𝑀𝑉𝐸𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
5 satfdm 33331 . . . 4 (((𝑀𝑉𝐸𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛))
64, 5syl 17 . . 3 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛))
7 fveq2 6774 . . . . . . 7 (𝑛 = 𝑁 → ((𝑀 Sat 𝐸)‘𝑛) = ((𝑀 Sat 𝐸)‘𝑁))
87dmeqd 5814 . . . . . 6 (𝑛 = 𝑁 → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑀 Sat 𝐸)‘𝑁))
9 fveq2 6774 . . . . . . 7 (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
109dmeqd 5814 . . . . . 6 (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
118, 10eqeq12d 2754 . . . . 5 (𝑛 = 𝑁 → (dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) ↔ dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
1211rspcv 3557 . . . 4 (𝑁 ∈ ω → (∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
13123ad2ant3 1134 . . 3 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
146, 13mpd 15 . 2 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
15 elelsuc 6338 . . . 4 (𝑁 ∈ ω → 𝑁 ∈ suc ω)
16153ad2ant3 1134 . . 3 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → 𝑁 ∈ suc ω)
17 fmlafv 33342 . . 3 (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
1816, 17syl 17 . 2 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
1914, 18eqtr4d 2781 1 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  Vcvv 3432  c0 4256  dom cdm 5589  suc csuc 6268  cfv 6433  (class class class)co 7275  ωcom 7712   Sat csat 33298  Fmlacfmla 33299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-goel 33302  df-goal 33304  df-sat 33305  df-fmla 33307
This theorem is referenced by:  satffunlem1lem2  33365  satffunlem2lem2  33368  satff  33372  satefvfmla0  33380  satefvfmla1  33387
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