Theorem satf00 35517

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 7829	. . 3 ⊢ ∅ ∈ ω
2		elelsuc 6390	. . 3 ⊢ (∅ ∈ ω → ∅ ∈ suc ω)
3		satf0sucom 35516	. . 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 9550	. . . 4 ⊢ ω ∈ V
6	5, 5	xpex 7696	. . . 4 ⊢ (ω × ω) ∈ V
7		xpexg 7693	. . . . 5 ⊢ ((ω ∈ V ∧ (ω × ω) ∈ V) → (ω × (ω × ω)) ∈ V)
8		simpl 482	. . . . 5 ⊢ ((ω ∈ V ∧ (ω × ω) ∈ V) → ω ∈ V)
9		goelel3xp 35491	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) ∈ (ω × (ω × ω)))
10		eleq1 2822	. . . . . . . 8 ⊢ (𝑥 = (𝑖∈𝑔𝑗) → (𝑥 ∈ (ω × (ω × ω)) ↔ (𝑖∈𝑔𝑗) ∈ (ω × (ω × ω))))
11	9, 10	syl5ibrcom 247	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖∈𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω))))
12	11	rexlimivv 3176	. . . . . 6 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω)))
13	12	ad2antll 729	. . . . 5 ⊢ (((ω ∈ V ∧ (ω × ω) ∈ V) ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))) → 𝑥 ∈ (ω × (ω × ω)))
14		eleq1 2822	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ∈ ω ↔ ∅ ∈ ω))
15	1, 14	mpbiri 258	. . . . . 6 ⊢ (𝑦 = ∅ → 𝑦 ∈ ω)
16	15	ad2antrl 728	. . . . 5 ⊢ (((ω ∈ V ∧ (ω × ω) ∈ V) ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))) → 𝑦 ∈ ω)
17	7, 8, 13, 16	opabex2 7999	. . . 4 ⊢ ((ω ∈ V ∧ (ω × ω) ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} ∈ V)
18	5, 6, 17	mp2an 692	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))} ∈ V
19	18	rdg0 8350	. 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 2757	1 ⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}

Syntax hints:   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  Vcvv 3438   ∪ cun 3897  ∅c0 4283  {copab 5158   ↦ cmpt 5177   × cxp 5620  suc csuc 6317  ‘cfv 6490  (class class class)co 7356  ωcom 7806  1^st c1st 7929  reccrdg 8338  ∈_𝑔cgoe 35476  ⊼_𝑔cgna 35477  ∀_𝑔cgol 35478   Sat csat 35479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-map 8763  df-goel 35483  df-sat 35486

This theorem is referenced by:  satf0op  35520  satf0n0  35521  sat1el2xp  35522  fmla0  35525  fmlafvel  35528