Theorem satfsschain 36296

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 6908	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑆‘𝑏) = (𝑆‘𝐵))
2	1	sseq2d 3945	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑆‘𝐵) ⊆ (𝑆‘𝑏) ↔ (𝑆‘𝐵) ⊆ (𝑆‘𝐵)))
3	2	imbi2d 343	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑏)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝐵))))
4		fveq2 6908	. . . . . . 7 ⊢ (𝑏 = 𝑎 → (𝑆‘𝑏) = (𝑆‘𝑎))
5	4	sseq2d 3945	. . . . . 6 ⊢ (𝑏 = 𝑎 → ((𝑆‘𝐵) ⊆ (𝑆‘𝑏) ↔ (𝑆‘𝐵) ⊆ (𝑆‘𝑎)))
6	5	imbi2d 343	. . . . 5 ⊢ (𝑏 = 𝑎 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑏)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎))))
7		fveq2 6908	. . . . . . 7 ⊢ (𝑏 = suc 𝑎 → (𝑆‘𝑏) = (𝑆‘suc 𝑎))
8	7	sseq2d 3945	. . . . . 6 ⊢ (𝑏 = suc 𝑎 → ((𝑆‘𝐵) ⊆ (𝑆‘𝑏) ↔ (𝑆‘𝐵) ⊆ (𝑆‘suc 𝑎)))
9	8	imbi2d 343	. . . . 5 ⊢ (𝑏 = suc 𝑎 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑏)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘suc 𝑎))))
10		fveq2 6908	. . . . . . 7 ⊢ (𝑏 = 𝐴 → (𝑆‘𝑏) = (𝑆‘𝐴))
11	10	sseq2d 3945	. . . . . 6 ⊢ (𝑏 = 𝐴 → ((𝑆‘𝐵) ⊆ (𝑆‘𝑏) ↔ (𝑆‘𝐵) ⊆ (𝑆‘𝐴)))
12	11	imbi2d 343	. . . . 5 ⊢ (𝑏 = 𝐴 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑏)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝐴))))
13		ssidd 3936	. . . . . 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 43	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎)))
16	15	adantl 487	. . . . . . . 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 490	. . . . . . . . . 10 ⊢ (((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → (𝑆‘𝐵) ⊆ (𝑆‘𝑎))
18		ssun1 4179	. . . . . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝑀 ∈ 𝑉)
20		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝐸 ∈ 𝑊)
21		simplll 787	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → 𝑎 ∈ ω)
22		satfsschain.s	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (𝑀 Sat 𝐸)
23	22	satfvsuc 36293	. . . . . . . . . . . . 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 1451	. . . . . . . . . . . 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 3956	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (𝑆‘𝑎) ⊆ (𝑆‘suc 𝑎))
26	25	adantr 486	. . . . . . . . . 10 ⊢ (((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → (𝑆‘𝑎) ⊆ (𝑆‘suc 𝑎))
27	17, 26	sstrd 3923	. . . . . . . . 9 ⊢ (((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ (𝑆‘𝐵) ⊆ (𝑆‘𝑎)) → (𝑆‘𝐵) ⊆ (𝑆‘suc 𝑎))
28	27	ex 418	. . . . . . . 8 ⊢ ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → ((𝑆‘𝐵) ⊆ (𝑆‘𝑎) → (𝑆‘𝐵) ⊆ (𝑆‘suc 𝑎)))
29	16, 28	syld 48	. . . . . . 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 418	. . . . . 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 87	. . . . 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 7933	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝐴)))
33	32	ex 418	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝐴 → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘𝐵) ⊆ (𝑆‘𝐴))))
34	33	com23 87	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝐵 ⊆ 𝐴 → (𝑆‘𝐵) ⊆ (𝑆‘𝐴))))
35	34	impcom 413	1 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 ⊆ 𝐴 → (𝑆‘𝐵) ⊆ (𝑆‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  {csn 4588  ⟨cop 4838  {copab 5170   ↾ cres 5657  suc csuc 6382  ‘cfv 6558  (class class class)co 7443  ωcom 7901  1^st c1st 8023  2^nd c2nd 8024   ↑_m cmap 8875  ⊼_𝑔cgna 36266  ∀_𝑔cgol 36267   Sat csat 36268

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-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669

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 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-sat 36275

This theorem is used by:  satfvsucsuc  36297  satffunlem2lem2  36338  satffunlem2  36340  satfun  36343