Theorem ressval3d 16619

Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.)

Hypotheses

Ref	Expression
ressval3d.r	⊢ 𝑅 = (𝑆 ↾s 𝐴)
ressval3d.b	⊢ 𝐵 = (Base‘𝑆)
ressval3d.e	⊢ 𝐸 = (Base‘ndx)
ressval3d.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
ressval3d.f	⊢ (𝜑 → Fun 𝑆)
ressval3d.d	⊢ (𝜑 → 𝐸 ∈ dom 𝑆)
ressval3d.u	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
ressval3d	⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Proof of Theorem ressval3d

Step	Hyp	Ref	Expression
1		ressval3d.u	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sspss 4005	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
3		dfpss3 3992	. . . . 5 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
4	3	orbi1i 911	. . . 4 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) ↔ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵))
5	2, 4	bitri 278	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵))
6		simplr 768	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ¬ 𝐵 ⊆ 𝐴)
7		ressval3d.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
8	7	adantl 485	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑆 ∈ 𝑉)
9		simpl 486	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ 𝐵)
10		ressval3d.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑆)
11	10	fvexi 6672	. . . . . . . . 9 ⊢ 𝐵 ∈ V
12	11	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
13		ssexg 5193	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
14	9, 12, 13	syl2an 598	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐴 ∈ V)
15		ressval3d.r	. . . . . . . 8 ⊢ 𝑅 = (𝑆 ↾s 𝐴)
16	15, 10	ressval2 16611	. . . . . . 7 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
17	6, 8, 14, 16	syl3anc 1368	. . . . . 6 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
18		ressval3d.e	. . . . . . . . . 10 ⊢ 𝐸 = (Base‘ndx)
19	18	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐸 = (Base‘ndx))
20		df-ss 3875	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
21	20	biimpi 219	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
22	21	eqcomd 2764	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (𝐴 ∩ 𝐵))
23	22	adantr 484	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐴 ∩ 𝐵))
24	23	adantr 484	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐴 = (𝐴 ∩ 𝐵))
25	19, 24	opeq12d 4771	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)
26	25	eqcomd 2764	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩ = ⟨𝐸, 𝐴⟩)
27	26	oveq2d 7166	. . . . . 6 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
28	17, 27	eqtrd 2793	. . . . 5 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
29	28	ex 416	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
30	15	a1i 11	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = (𝑆 ↾s 𝐴))
31		oveq2 7158	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑆 ↾s 𝐴) = (𝑆 ↾s 𝐵))
32	31	adantr 484	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 ↾s 𝐴) = (𝑆 ↾s 𝐵))
33	7	adantl 485	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑆 ∈ 𝑉)
34	10	ressid 16617	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ↾s 𝐵) = 𝑆)
35	33, 34	syl 17	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 ↾s 𝐵) = 𝑆)
36	30, 32, 35	3eqtrd 2797	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = 𝑆)
37		df-base 16547	. . . . . . . 8 ⊢ Base = Slot 1
38		1nn 11685	. . . . . . . 8 ⊢ 1 ∈ ℕ
39		ressval3d.f	. . . . . . . 8 ⊢ (𝜑 → Fun 𝑆)
40		ressval3d.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ dom 𝑆)
41	18, 40	eqeltrrid 2857	. . . . . . . 8 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)
42	37, 38, 7, 39, 41	setsidvald 16572	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
43	42	adantl 485	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
44	18	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐸 = (Base‘ndx))
45		simpl 486	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐴 = 𝐵)
46	45, 10	eqtrdi 2809	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐴 = (Base‘𝑆))
47	44, 46	opeq12d 4771	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (Base‘𝑆)⟩)
48	47	eqcomd 2764	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → ⟨(Base‘ndx), (Base‘𝑆)⟩ = ⟨𝐸, 𝐴⟩)
49	48	oveq2d 7166	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
50	36, 43, 49	3eqtrd 2797	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
51	50	ex 416	. . . 4 ⊢ (𝐴 = 𝐵 → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
52	29, 51	jaoi 854	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵) → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
53	5, 52	sylbi 220	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
54	1, 53	mpcom 38	1 ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∩ cin 3857   ⊆ wss 3858   ⊊ wpss 3859  ⟨cop 4528  dom cdm 5524  Fun wfun 6329  ‘cfv 6335  (class class class)co 7150  1c1 10576  ndxcnx 16538   sSet csts 16539  Basecbs 16541   ↾_s cress 16542

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-1cn 10633  ax-addcl 10635

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-nn 11675  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549

This theorem is referenced by:  estrres  17455