Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  satf00 Structured version   Visualization version   GIF version

Theorem satf00 32629
 Description: The value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at ∅. (Contributed by AV, 14-Sep-2023.)
Assertion
Ref Expression
satf00 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
Distinct variable group:   𝑖,𝑗,𝑥,𝑦

Proof of Theorem satf00
Dummy variables 𝑓 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 peano1 7576 . . 3 ∅ ∈ ω
2 elelsuc 6236 . . 3 (∅ ∈ ω → ∅ ∈ suc ω)
3 satf0sucom 32628 . . 3 (∅ ∈ suc ω → ((∅ 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))})‘∅))
41, 2, 3mp2b 10 . 2 ((∅ 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))})‘∅)
5 omex 9082 . . . 4 ω ∈ V
65, 5xpex 7451 . . . 4 (ω × ω) ∈ V
7 xpexg 7448 . . . . 5 ((ω ∈ V ∧ (ω × ω) ∈ V) → (ω × (ω × ω)) ∈ V)
8 simpl 486 . . . . 5 ((ω ∈ V ∧ (ω × ω) ∈ V) → ω ∈ V)
9 goelel3xp 32603 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) ∈ (ω × (ω × ω)))
10 eleq1 2899 . . . . . . . 8 (𝑥 = (𝑖𝑔𝑗) → (𝑥 ∈ (ω × (ω × ω)) ↔ (𝑖𝑔𝑗) ∈ (ω × (ω × ω))))
119, 10syl5ibrcom 250 . . . . . . 7 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω))))
1211rexlimivv 3278 . . . . . 6 (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω)))
1312ad2antll 728 . . . . 5 (((ω ∈ V ∧ (ω × ω) ∈ V) ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → 𝑥 ∈ (ω × (ω × ω)))
14 eleq1 2899 . . . . . . 7 (𝑦 = ∅ → (𝑦 ∈ ω ↔ ∅ ∈ ω))
151, 14mpbiri 261 . . . . . 6 (𝑦 = ∅ → 𝑦 ∈ ω)
1615ad2antrl 727 . . . . 5 (((ω ∈ V ∧ (ω × ω) ∈ V) ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → 𝑦 ∈ ω)
177, 8, 13, 16opabex2 7730 . . . 4 ((ω ∈ V ∧ (ω × ω) ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} ∈ V)
185, 6, 17mp2an 691 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} ∈ V
1918rdg0 8032 . 2 (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))})‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
204, 19eqtri 2844 1 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2115  ∃wrex 3127  Vcvv 3471   ∪ cun 3908  ∅c0 4266  {copab 5101   ↦ cmpt 5119   × cxp 5526  suc csuc 6166  ‘cfv 6328  (class class class)co 7130  ωcom 7555  1st c1st 7662  reccrdg 8020  ∈𝑔cgoe 32588  ⊼𝑔cgna 32589  ∀𝑔cgol 32590   Sat csat 32591 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-map 8383  df-goel 32595  df-sat 32598 This theorem is referenced by:  satf0op  32632  satf0n0  32633  sat1el2xp  32634  fmla0  32637  fmlafvel  32640
 Copyright terms: Public domain W3C validator