Theorem satfsschain 34844

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	⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 ⊆ 𝐴 → (𝑆‘𝐵) ⊆ (𝑆‘𝐴)))

Proof of Theorem satfsschain

Dummy variables 𝑎 𝑏 𝑖 𝑘 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑆‘𝑏) = (𝑆‘𝐵))
2	1	sseq2d 4006	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑆‘𝐵) ⊆ (𝑆‘𝑏) ↔ (𝑆‘𝐵) ⊆ (𝑆‘𝐵)))
3	2	imbi2d 340	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑏)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝐵))))
4		fveq2 6881	. . . . . . 7 ⊢ (𝑏 = 𝑎 → (𝑆‘𝑏) = (𝑆‘𝑎))
5	4	sseq2d 4006	. . . . . 6 ⊢ (𝑏 = 𝑎 → ((𝑆‘𝐵) ⊆ (𝑆‘𝑏) ↔ (𝑆‘𝐵) ⊆ (𝑆‘𝑎)))
6	5	imbi2d 340	. . . . 5 ⊢ (𝑏 = 𝑎 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑏)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎))))
7		fveq2 6881	. . . . . . 7 ⊢ (𝑏 = suc 𝑎 → (𝑆‘𝑏) = (𝑆‘suc 𝑎))
8	7	sseq2d 4006	. . . . . 6 ⊢ (𝑏 = suc 𝑎 → ((𝑆‘𝐵) ⊆ (𝑆‘𝑏) ↔ (𝑆‘𝐵) ⊆ (𝑆‘suc 𝑎)))
9	8	imbi2d 340	. . . . 5 ⊢ (𝑏 = suc 𝑎 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑏)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘suc 𝑎))))
10		fveq2 6881	. . . . . . 7 ⊢ (𝑏 = 𝐴 → (𝑆‘𝑏) = (𝑆‘𝐴))
11	10	sseq2d 4006	. . . . . 6 ⊢ (𝑏 = 𝐴 → ((𝑆‘𝐵) ⊆ (𝑆‘𝑏) ↔ (𝑆‘𝐵) ⊆ (𝑆‘𝐴)))
12	11	imbi2d 340	. . . . 5 ⊢ (𝑏 = 𝐴 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑏)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝐴))))
13		ssidd 3997	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝐵))
14	13	a1i 11	. . . . 5 ⊢ (𝐵 ∈ ω → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝐵)))
15		pm2.27 42	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎)))
16	15	adantl 481	. . . . . . . 8 ⊢ ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎)))
17		simpr 484	. . . . . . . . . 10 ⊢ (((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎))
18		ssun1 4164	. . . . . . . . . . . 12 ⊢ (𝑆‘𝑎) ⊆ ((𝑆‘𝑎) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑎)(∃𝑣 ∈ (𝑆‘𝑎)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑧 ∈ (𝑀 ↑m ω) ∣ ∀𝑘 ∈ 𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑧 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})
19		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝑀 ∈ 𝑉)
20		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝐸 ∈ 𝑊)
21		simplll 772	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → 𝑎 ∈ ω)
22		satfsschain.s	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (𝑀 Sat 𝐸)
23	22	satfvsuc 34841	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑎 ∈ ω) → (𝑆‘suc 𝑎) = ((𝑆‘𝑎) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑎)(∃𝑣 ∈ (𝑆‘𝑎)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑧 ∈ (𝑀 ↑m ω) ∣ ∀𝑘 ∈ 𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑧 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
24	19, 20, 21, 23	syl2an23an 1420	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (𝑆‘suc 𝑎) = ((𝑆‘𝑎) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑎)(∃𝑣 ∈ (𝑆‘𝑎)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑧 ∈ (𝑀 ↑m ω) ∣ ∀𝑘 ∈ 𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑧 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
25	18, 24	sseqtrrid 4027	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (𝑆‘𝑎) ⊆ (𝑆‘suc 𝑎))
26	25	adantr 480	. . . . . . . . . 10 ⊢ (((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → (𝑆‘𝑎) ⊆ (𝑆‘suc 𝑎))
27	17, 26	sstrd 3984	. . . . . . . . 9 ⊢ (((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → (𝑆‘𝐵) ⊆ (𝑆‘suc 𝑎))
28	27	ex 412	. . . . . . . 8 ⊢ ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → ((𝑆‘𝐵) ⊆ (𝑆‘𝑎) → (𝑆‘𝐵) ⊆ (𝑆‘suc 𝑎)))
29	16, 28	syld 47	. . . . . . 7 ⊢ ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → (𝑆‘𝐵) ⊆ (𝑆‘suc 𝑎)))
30	29	ex 412	. . . . . 6 ⊢ (((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → (𝑆‘𝐵) ⊆ (𝑆‘suc 𝑎))))
31	30	com23 86	. . . . 5 ⊢ (((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘suc 𝑎))))
32	3, 6, 9, 12, 14, 31	findsg 7883	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝐴)))
33	32	ex 412	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝐴 → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝐴))))
34	33	com23 86	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝐵 ⊆ 𝐴 → (𝑆‘𝐵) ⊆ (𝑆‘𝐴))))
35	34	impcom 407	1 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 ⊆ 𝐴 → (𝑆‘𝐵) ⊆ (𝑆‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098  ∀wral 3053  ∃wrex 3062  {crab 3424   ∖ cdif 3937   ∪ cun 3938   ∩ cin 3939   ⊆ wss 3940  {csn 4620  ⟨cop 4626  {copab 5200   ↾ cres 5668  suc csuc 6356  ‘cfv 6533  (class class class)co 7401  ωcom 7848  1^st c1st 7966  2^nd c2nd 7967   ↑_m cmap 8816  ⊼_𝑔cgna 34814  ∀_𝑔cgol 34815   Sat csat 34816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9632

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-sat 34823

This theorem is referenced by:  satfvsucsuc  34845  satffunlem2lem2  34886  satffunlem2  34888  satfun  34891