Theorem satfsucom 36285

Description: The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at an element of the successor of ω. (Contributed by AV, 22-Sep-2023.)

Assertion

Ref	Expression
satfsucom	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ suc ω) → ((𝑀 Sat 𝐸)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘𝑁))

Distinct variable groups:   𝐸,𝑎,𝑓,𝑖,𝑗,𝑢,𝑣,𝑥,𝑦   𝑀,𝑎,𝑓,𝑖,𝑗,𝑢,𝑣,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐸(𝑧)   𝑁(𝑥, 𝑦, 𝑧, 𝑣, 𝑢, 𝑓, 𝑖, 𝑗, 𝑎)   𝑉(𝑥, 𝑦, 𝑧, 𝑣, 𝑢, 𝑓, 𝑖, 𝑗, 𝑎)   𝑊(𝑥, 𝑦, 𝑧, 𝑣, 𝑢, 𝑓, 𝑖, 𝑗, 𝑎)

Proof of Theorem satfsucom

Step	Hyp	Ref	Expression
1		satf 36284	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω))
2	1	fveq1d 6907	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘𝑁) = ((rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω)‘𝑁))
3	2	3adant3 1150	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ suc ω) → ((𝑀 Sat 𝐸)‘𝑁) = ((rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω)‘𝑁))
4		fvres 6924	. . 3 ⊢ (𝑁 ∈ suc ω → ((rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘𝑁))
5	4	3ad2ant3 1153	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ suc ω) → ((rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘𝑁))
6	3, 5	eqtrd 2798	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ suc ω) → ((𝑀 Sat 𝐸)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  {csn 4588  ⟨cop 4838   class class class wbr 5106  {copab 5170   ↦ cmpt 5189   ↾ cres 5657  suc csuc 6379  ‘cfv 6555  (class class class)co 7440  ωcom 7898  1^st c1st 8020  2^nd c2nd 8021  reccrdg 8442   ↑_m cmap 8872  ∈_𝑔cgoe 36264  ⊼_𝑔cgna 36265  ∀_𝑔cgol 36266   Sat csat 36267

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772  ax-inf2 9666

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-sat 36274

This theorem is used by: (None)