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Theorem satfdmfmla 32933
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 5175 . . . . . . 7 ∅ ∈ V
21, 1pm3.2i 474 . . . . . 6 (∅ ∈ V ∧ ∅ ∈ V)
32jctr 528 . . . . 5 ((𝑀𝑉𝐸𝑊) → ((𝑀𝑉𝐸𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
433adant3 1133 . . . 4 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → ((𝑀𝑉𝐸𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
5 satfdm 32902 . . . 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 6674 . . . . . . 7 (𝑛 = 𝑁 → ((𝑀 Sat 𝐸)‘𝑛) = ((𝑀 Sat 𝐸)‘𝑁))
87dmeqd 5748 . . . . . 6 (𝑛 = 𝑁 → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑀 Sat 𝐸)‘𝑁))
9 fveq2 6674 . . . . . . 7 (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
109dmeqd 5748 . . . . . 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 3521 . . . 4 (𝑁 ∈ ω → (∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
13123ad2ant3 1136 . . 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 6244 . . . 4 (𝑁 ∈ ω → 𝑁 ∈ suc ω)
16153ad2ant3 1136 . . 3 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → 𝑁 ∈ suc ω)
17 fmlafv 32913 . . 3 (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
1816, 17syl 17 . 2 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
1914, 18eqtr4d 2776 1 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  Vcvv 3398  c0 4211  dom cdm 5525  suc csuc 6174  cfv 6339  (class class class)co 7170  ωcom 7599   Sat csat 32869  Fmlacfmla 32870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-goel 32873  df-goal 32875  df-sat 32876  df-fmla 32878
This theorem is referenced by:  satffunlem1lem2  32936  satffunlem2lem2  32939  satff  32943  satefvfmla0  32951  satefvfmla1  32958
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