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Theorem satfdmfmla 32642
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 5203 . . . . . . 7 ∅ ∈ V
21, 1pm3.2i 473 . . . . . 6 (∅ ∈ V ∧ ∅ ∈ V)
32jctr 527 . . . . 5 ((𝑀𝑉𝐸𝑊) → ((𝑀𝑉𝐸𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
433adant3 1128 . . . 4 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → ((𝑀𝑉𝐸𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
5 satfdm 32611 . . . 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 6664 . . . . . . 7 (𝑛 = 𝑁 → ((𝑀 Sat 𝐸)‘𝑛) = ((𝑀 Sat 𝐸)‘𝑁))
87dmeqd 5768 . . . . . 6 (𝑛 = 𝑁 → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑀 Sat 𝐸)‘𝑁))
9 fveq2 6664 . . . . . . 7 (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
109dmeqd 5768 . . . . . 6 (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
118, 10eqeq12d 2837 . . . . 5 (𝑛 = 𝑁 → (dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) ↔ dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
1211rspcv 3617 . . . 4 (𝑁 ∈ ω → (∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
13123ad2ant3 1131 . . 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 6257 . . . 4 (𝑁 ∈ ω → 𝑁 ∈ suc ω)
16153ad2ant3 1131 . . 3 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → 𝑁 ∈ suc ω)
17 fmlafv 32622 . . 3 (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
1816, 17syl 17 . 2 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
1914, 18eqtr4d 2859 1 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138  Vcvv 3494  c0 4290  dom cdm 5549  suc csuc 6187  cfv 6349  (class class class)co 7150  ωcom 7574   Sat csat 32578  Fmlacfmla 32579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-goel 32582  df-goal 32584  df-sat 32585  df-fmla 32587
This theorem is referenced by:  satffunlem1lem2  32645  satffunlem2lem2  32648  satff  32652  satefvfmla0  32660  satefvfmla1  32667
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