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Theorem satf0sucom 32909
Description: The satisfaction predicate as function over wff codes in the empty model with an empty binary relation at a successor of ω. (Contributed by AV, 14-Sep-2023.)
Assertion
Ref Expression
satf0sucom (𝑁 ∈ suc ω → ((∅ Sat ∅)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑁))
Distinct variable group:   𝑓,𝑖,𝑗,𝑢,𝑣,𝑥,𝑦
Allowed substitution hints:   𝑁(𝑥,𝑦,𝑣,𝑢,𝑓,𝑖,𝑗)

Proof of Theorem satf0sucom
StepHypRef Expression
1 satf0 32908 . . 3 (∅ Sat ∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}) ↾ suc ω)
21fveq1i 6678 . 2 ((∅ Sat ∅)‘𝑁) = ((rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}) ↾ suc ω)‘𝑁)
3 fvres 6696 . 2 (𝑁 ∈ suc ω → ((rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}) ↾ suc ω)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑁))
42, 3syl5eq 2786 1 (𝑁 ∈ suc ω → ((∅ Sat ∅)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 846   = wceq 1542  wcel 2114  wrex 3055  Vcvv 3399  cun 3842  c0 4212  {copab 5093  cmpt 5111  cres 5528  suc csuc 6175  cfv 6340  (class class class)co 7173  ωcom 7602  1st c1st 7715  reccrdg 8077  𝑔cgoe 32869  𝑔cgna 32870  𝑔cgol 32871   Sat csat 32872
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 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482  ax-inf2 9180
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 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-om 7603  df-1st 7717  df-2nd 7718  df-wrecs 7979  df-recs 8040  df-rdg 8078  df-map 8442  df-sat 32879
This theorem is referenced by:  satf00  32910  satf0suc  32912  sat1el2xp  32915  fmla  32917  fmlasuc0  32920
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