Theorem satfsschain 35577

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 6842	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑆‘𝑏) = (𝑆‘𝐵))
2	1	sseq2d 3968	. . . . . 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 6842	. . . . . . 7 ⊢ (𝑏 = 𝑎 → (𝑆‘𝑏) = (𝑆‘𝑎))
5	4	sseq2d 3968	. . . . . 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 6842	. . . . . . 7 ⊢ (𝑏 = suc 𝑎 → (𝑆‘𝑏) = (𝑆‘suc 𝑎))
8	7	sseq2d 3968	. . . . . 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 6842	. . . . . . 7 ⊢ (𝑏 = 𝐴 → (𝑆‘𝑏) = (𝑆‘𝐴))
11	10	sseq2d 3968	. . . . . 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 3959	. . . . . 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 4132	. . . . . . . . . . . 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 775	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → 𝑎 ∈ ω)
22		satfsschain.s	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (𝑀 Sat 𝐸)
23	22	satfvsuc 35574	. . . . . . . . . . . . 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 1426	. . . . . . . . . . . 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 3979	. . . . . . . . . . 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 3946	. . . . . . . . 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 7849	. . . 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 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  {csn 4582  ⟨cop 4588  {copab 5162   ↾ cres 5634  suc csuc 6327  ‘cfv 6500  (class class class)co 7368  ωcom 7818  1^st c1st 7941  2^nd c2nd 7942   ↑_m cmap 8775  ⊼_𝑔cgna 35547  ∀_𝑔cgol 35548   Sat csat 35549

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-sat 35556

This theorem is referenced by:  satfvsucsuc  35578  satffunlem2lem2  35619  satffunlem2  35621  satfun  35624