Theorem satf00 36229

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 = (𝑖∈𝑔𝑗))}

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.

Step	Hyp	Ref	Expression
1		peano1 7904	. . 3 ⊢ ∅ ∈ ω
2		elelsuc 6441	. . 3 ⊢ (∅ ∈ ω → ∅ ∈ suc ω)
3		satf0sucom 36228	. . 3 ⊢ (∅ ∈ suc ω → ((∅ Sat ∅)‘∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘∅))
4	1, 2, 3	mp2b 10	. 2 ⊢ ((∅ Sat ∅)‘∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘∅)
5		omex 9651	. . . 4 ⊢ ω ∈ V
6	5, 5	xpex 7771	. . . 4 ⊢ (ω × ω) ∈ V
7		xpexg 7768	. . . . 5 ⊢ ((ω ∈ V ∧ (ω × ω) ∈ V) → (ω × (ω × ω)) ∈ V)
8		simpl 488	. . . . 5 ⊢ ((ω ∈ V ∧ (ω × ω) ∈ V) → ω ∈ V)
9		goelel3xp 36203	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) ∈ (ω × (ω × ω)))
10		eleq1 2848	. . . . . . . 8 ⊢ (𝑥 = (𝑖∈𝑔𝑗) → (𝑥 ∈ (ω × (ω × ω)) ↔ (𝑖∈𝑔𝑗) ∈ (ω × (ω × ω))))
11	9, 10	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖∈𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω))))
12	11	rexlimivv 3204	. . . . . 6 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω)))
13	12	ad2antll 742	. . . . 5 ⊢ (((ω ∈ V ∧ (ω × ω) ∈ V) ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))) → 𝑥 ∈ (ω × (ω × ω)))
14		eleq1 2848	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ∈ ω ↔ ∅ ∈ ω))
15	1, 14	mpbiri 261	. . . . . 6 ⊢ (𝑦 = ∅ → 𝑦 ∈ ω)
16	15	ad2antrl 741	. . . . 5 ⊢ (((ω ∈ V ∧ (ω × ω) ∈ V) ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))) → 𝑦 ∈ ω)
17	7, 8, 13, 16	opabex2 8072	. . . 4 ⊢ ((ω ∈ V ∧ (ω × ω) ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} ∈ V)
18	5, 6, 17	mp2an 705	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} ∈ V
19	18	rdg0 8436	. 2 ⊢ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
20	4, 19	eqtri 2783	1 ⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}

This proof depends on syntax axioms:   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450   ∪ cun 4148  ∅c0 4277  {copab 5166   ↦ cmpt 5185   × cxp 5649  suc csuc 6367  ‘cfv 6543  (class class class)co 7424  ωcom 7881  1^st c1st 8003  reccrdg 8424  ∈_𝑔cgoe 36188  ⊼_𝑔cgna 36189  ∀_𝑔cgol 36190   Sat csat 36191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-map 8856  df-goel 36195  df-sat 36198

This theorem is used by:  satf0op  36232  satf0n0  36233  sat1el2xp  36234  fmla0  36237  fmlafvel  36240