Theorem ressval3d 17306

Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.) (Proof shortened by AV, 17-Oct-2024.)

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 4064	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
3		dfpss3 4051	. . . . 5 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
4	3	orbi1i 926	. . . 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 780	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ¬ 𝐵 ⊆ 𝐴)
7		ressval3d.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
8	7	adantl 486	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑆 ∈ 𝑉)
9		simpl 487	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ 𝐵)
10		ressval3d.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑆)
11	10	fvexi 6896	. . . . . . . . 9 ⊢ 𝐵 ∈ V
12	11	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
13		ssexg 5294	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
14	9, 12, 13	syl2an 607	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐴 ∈ V)
15		ressval3d.r	. . . . . . . 8 ⊢ 𝑅 = (𝑆 ↾s 𝐴)
16	15, 10	ressval2 17295	. . . . . . 7 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
17	6, 8, 14, 16	syl3anc 1396	. . . . . 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		dfss2 3931	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
21	20	biimpi 219	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
22	21	eqcomd 2775	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (𝐴 ∩ 𝐵))
23	22	adantr 485	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐴 ∩ 𝐵))
24	23	adantr 485	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐴 = (𝐴 ∩ 𝐵))
25	19, 24	opeq12d 4850	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)
26	25	eqcomd 2775	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩ = ⟨𝐸, 𝐴⟩)
27	26	oveq2d 7427	. . . . . 6 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
28	17, 27	eqtrd 2804	. . . . 5 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
29	28	ex 417	. . . 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 7419	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑆 ↾s 𝐴) = (𝑆 ↾s 𝐵))
32	31	adantr 485	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 ↾s 𝐴) = (𝑆 ↾s 𝐵))
33	7	adantl 486	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑆 ∈ 𝑉)
34	10	ressid 17304	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ↾s 𝐵) = 𝑆)
35	33, 34	syl 18	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 ↾s 𝐵) = 𝑆)
36	30, 32, 35	3eqtrd 2808	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = 𝑆)
37		baseid 17272	. . . . . . . 8 ⊢ Base = Slot (Base‘ndx)
38		ressval3d.f	. . . . . . . 8 ⊢ (𝜑 → Fun 𝑆)
39		ressval3d.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ dom 𝑆)
40	18, 39	eqeltrrid 2874	. . . . . . . 8 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)
41	37, 7, 38, 40	setsidvald 17259	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
42	41	adantl 486	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
43	18	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐸 = (Base‘ndx))
44		simpl 487	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐴 = 𝐵)
45	44, 10	eqtrdi 2820	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐴 = (Base‘𝑆))
46	43, 45	opeq12d 4850	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (Base‘𝑆)⟩)
47	46	eqcomd 2775	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → ⟨(Base‘ndx), (Base‘𝑆)⟩ = ⟨𝐸, 𝐴⟩)
48	47	oveq2d 7427	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
49	36, 42, 48	3eqtrd 2808	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
50	49	ex 417	. . . 4 ⊢ (𝐴 = 𝐵 → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
51	29, 50	jaoi 870	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵) → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
52	5, 51	sylbi 220	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
53	1, 52	mpcom 39	1 ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913   ⊊ wpss 3914  ⟨cop 4600  dom cdm 5662  Fun wfun 6531  ‘cfv 6537  (class class class)co 7411   sSet csts 17223  ndxcnx 17253  Basecbs 17269   ↾_s cress 17290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-1cn 11158  ax-addcl 11160

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-nn 12234  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291

This theorem is referenced by:  estrres  18195