Theorem satfsschain 35376

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 6817	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑆‘𝑏) = (𝑆‘𝐵))
2	1	sseq2d 3965	. . . . . 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 6817	. . . . . . 7 ⊢ (𝑏 = 𝑎 → (𝑆‘𝑏) = (𝑆‘𝑎))
5	4	sseq2d 3965	. . . . . 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 6817	. . . . . . 7 ⊢ (𝑏 = suc 𝑎 → (𝑆‘𝑏) = (𝑆‘suc 𝑎))
8	7	sseq2d 3965	. . . . . 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 6817	. . . . . . 7 ⊢ (𝑏 = 𝐴 → (𝑆‘𝑏) = (𝑆‘𝐴))
11	10	sseq2d 3965	. . . . . 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 3956	. . . . . 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 4126	. . . . . . . . . . . 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 774	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → 𝑎 ∈ ω)
22		satfsschain.s	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (𝑀 Sat 𝐸)
23	22	satfvsuc 35373	. . . . . . . . . . . . 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 1425	. . . . . . . . . . . 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 3976	. . . . . . . . . . 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 3943	. . . . . . . . 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 7822	. . . 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 847   = wceq 1541   ∈ wcel 2110  ∀wral 3045  ∃wrex 3054  {crab 3393   ∖ cdif 3897   ∪ cun 3898   ∩ cin 3899   ⊆ wss 3900  {csn 4574  ⟨cop 4580  {copab 5151   ↾ cres 5616  suc csuc 6304  ‘cfv 6477  (class class class)co 7341  ωcom 7791  1^st c1st 7914  2^nd c2nd 7915   ↑_m cmap 8745  ⊼_𝑔cgna 35346  ∀_𝑔cgol 35347   Sat csat 35348

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-inf2 9526

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-sat 35355

This theorem is referenced by:  satfvsucsuc  35377  satffunlem2lem2  35418  satffunlem2  35420  satfun  35423