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Theorem satfsschain 32629
 Description: The binary relation of a satisfaction predicate as function over wff codes is an increasing chain (with respect to inclusion). (Contributed by AV, 15-Oct-2023.)
Hypothesis
Ref Expression
satfsschain.s 𝑆 = (𝑀 Sat 𝐸)
Assertion
Ref Expression
satfsschain (((𝑀𝑉𝐸𝑊) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵𝐴 → (𝑆𝐵) ⊆ (𝑆𝐴)))

Proof of Theorem satfsschain
Dummy variables 𝑎 𝑏 𝑖 𝑘 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6651 . . . . . . 7 (𝑏 = 𝐵 → (𝑆𝑏) = (𝑆𝐵))
21sseq2d 3983 . . . . . 6 (𝑏 = 𝐵 → ((𝑆𝐵) ⊆ (𝑆𝑏) ↔ (𝑆𝐵) ⊆ (𝑆𝐵)))
32imbi2d 344 . . . . 5 (𝑏 = 𝐵 → (((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝑏)) ↔ ((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝐵))))
4 fveq2 6651 . . . . . . 7 (𝑏 = 𝑎 → (𝑆𝑏) = (𝑆𝑎))
54sseq2d 3983 . . . . . 6 (𝑏 = 𝑎 → ((𝑆𝐵) ⊆ (𝑆𝑏) ↔ (𝑆𝐵) ⊆ (𝑆𝑎)))
65imbi2d 344 . . . . 5 (𝑏 = 𝑎 → (((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝑏)) ↔ ((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝑎))))
7 fveq2 6651 . . . . . . 7 (𝑏 = suc 𝑎 → (𝑆𝑏) = (𝑆‘suc 𝑎))
87sseq2d 3983 . . . . . 6 (𝑏 = suc 𝑎 → ((𝑆𝐵) ⊆ (𝑆𝑏) ↔ (𝑆𝐵) ⊆ (𝑆‘suc 𝑎)))
98imbi2d 344 . . . . 5 (𝑏 = suc 𝑎 → (((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝑏)) ↔ ((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆‘suc 𝑎))))
10 fveq2 6651 . . . . . . 7 (𝑏 = 𝐴 → (𝑆𝑏) = (𝑆𝐴))
1110sseq2d 3983 . . . . . 6 (𝑏 = 𝐴 → ((𝑆𝐵) ⊆ (𝑆𝑏) ↔ (𝑆𝐵) ⊆ (𝑆𝐴)))
1211imbi2d 344 . . . . 5 (𝑏 = 𝐴 → (((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝑏)) ↔ ((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝐴))))
13 ssidd 3974 . . . . . 6 ((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝐵))
1413a1i 11 . . . . 5 (𝐵 ∈ ω → ((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝐵)))
15 pm2.27 42 . . . . . . . . 9 ((𝑀𝑉𝐸𝑊) → (((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝑎)) → (𝑆𝐵) ⊆ (𝑆𝑎)))
1615adantl 485 . . . . . . . 8 ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑎) ∧ (𝑀𝑉𝐸𝑊)) → (((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝑎)) → (𝑆𝐵) ⊆ (𝑆𝑎)))
17 simpr 488 . . . . . . . . . 10 (((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑎) ∧ (𝑀𝑉𝐸𝑊)) ∧ (𝑆𝐵) ⊆ (𝑆𝑎)) → (𝑆𝐵) ⊆ (𝑆𝑎))
18 ssun1 4132 . . . . . . . . . . . 12 (𝑆𝑎) ⊆ ((𝑆𝑎) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑎)(∃𝑣 ∈ (𝑆𝑎)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑧 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑧 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})
19 simpl 486 . . . . . . . . . . . . 13 ((𝑀𝑉𝐸𝑊) → 𝑀𝑉)
20 simpr 488 . . . . . . . . . . . . 13 ((𝑀𝑉𝐸𝑊) → 𝐸𝑊)
21 simplll 774 . . . . . . . . . . . . 13 ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑎) ∧ (𝑀𝑉𝐸𝑊)) → 𝑎 ∈ ω)
22 satfsschain.s . . . . . . . . . . . . . 14 𝑆 = (𝑀 Sat 𝐸)
2322satfvsuc 32626 . . . . . . . . . . . . 13 ((𝑀𝑉𝐸𝑊𝑎 ∈ ω) → (𝑆‘suc 𝑎) = ((𝑆𝑎) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑎)(∃𝑣 ∈ (𝑆𝑎)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑧 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑧 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}))
2419, 20, 21, 23syl2an23an 1420 . . . . . . . . . . . 12 ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑎) ∧ (𝑀𝑉𝐸𝑊)) → (𝑆‘suc 𝑎) = ((𝑆𝑎) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑎)(∃𝑣 ∈ (𝑆𝑎)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑧 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑧 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}))
2518, 24sseqtrrid 4004 . . . . . . . . . . 11 ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑎) ∧ (𝑀𝑉𝐸𝑊)) → (𝑆𝑎) ⊆ (𝑆‘suc 𝑎))
2625adantr 484 . . . . . . . . . 10 (((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑎) ∧ (𝑀𝑉𝐸𝑊)) ∧ (𝑆𝐵) ⊆ (𝑆𝑎)) → (𝑆𝑎) ⊆ (𝑆‘suc 𝑎))
2717, 26sstrd 3961 . . . . . . . . 9 (((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑎) ∧ (𝑀𝑉𝐸𝑊)) ∧ (𝑆𝐵) ⊆ (𝑆𝑎)) → (𝑆𝐵) ⊆ (𝑆‘suc 𝑎))
2827ex 416 . . . . . . . 8 ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑎) ∧ (𝑀𝑉𝐸𝑊)) → ((𝑆𝐵) ⊆ (𝑆𝑎) → (𝑆𝐵) ⊆ (𝑆‘suc 𝑎)))
2916, 28syld 47 . . . . . . 7 ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑎) ∧ (𝑀𝑉𝐸𝑊)) → (((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝑎)) → (𝑆𝐵) ⊆ (𝑆‘suc 𝑎)))
3029ex 416 . . . . . 6 (((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑎) → ((𝑀𝑉𝐸𝑊) → (((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝑎)) → (𝑆𝐵) ⊆ (𝑆‘suc 𝑎))))
3130com23 86 . . . . 5 (((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑎) → (((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝑎)) → ((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆‘suc 𝑎))))
323, 6, 9, 12, 14, 31findsg 7592 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → ((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝐴)))
3332ex 416 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵𝐴 → ((𝑀𝑉𝐸𝑊) → (𝑆𝐵) ⊆ (𝑆𝐴))))
3433com23 86 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑀𝑉𝐸𝑊) → (𝐵𝐴 → (𝑆𝐵) ⊆ (𝑆𝐴))))
3534impcom 411 1 (((𝑀𝑉𝐸𝑊) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵𝐴 → (𝑆𝐵) ⊆ (𝑆𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2115  ∀wral 3132  ∃wrex 3133  {crab 3136   ∖ cdif 3915   ∪ cun 3916   ∩ cin 3917   ⊆ wss 3918  {csn 4548  ⟨cop 4554  {copab 5109   ↾ cres 5538  suc csuc 6174  ‘cfv 6336  (class class class)co 7138  ωcom 7563  1st c1st 7670  2nd c2nd 7671   ↑m cmap 8389  ⊼𝑔cgna 32599  ∀𝑔cgol 32600   Sat csat 32601 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 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-inf2 9088 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 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-sat 32608 This theorem is referenced by:  satfvsucsuc  32630  satffunlem2lem2  32671  satffunlem2  32673  satfun  32676
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