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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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