Theorem ressval3d 16563

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 4078	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
3		dfpss3 4065	. . . . 5 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
4	3	orbi1i 910	. . . 4 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) ↔ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵))
5	2, 4	bitri 277	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵))
6		simplr 767	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ¬ 𝐵 ⊆ 𝐴)
7		ressval3d.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
8	7	adantl 484	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑆 ∈ 𝑉)
9		simpl 485	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ 𝐵)
10		ressval3d.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑆)
11	10	fvexi 6686	. . . . . . . . 9 ⊢ 𝐵 ∈ V
12	11	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
13		ssexg 5229	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
14	9, 12, 13	syl2an 597	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐴 ∈ V)
15		ressval3d.r	. . . . . . . 8 ⊢ 𝑅 = (𝑆 ↾s 𝐴)
16	15, 10	ressval2 16555	. . . . . . 7 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
17	6, 8, 14, 16	syl3anc 1367	. . . . . 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 3954	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
21	20	biimpi 218	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
22	21	eqcomd 2829	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (𝐴 ∩ 𝐵))
23	22	adantr 483	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐴 ∩ 𝐵))
24	23	adantr 483	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐴 = (𝐴 ∩ 𝐵))
25	19, 24	opeq12d 4813	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)
26	25	eqcomd 2829	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩ = ⟨𝐸, 𝐴⟩)
27	26	oveq2d 7174	. . . . . 6 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
28	17, 27	eqtrd 2858	. . . . 5 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
29	28	ex 415	. . . 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 7166	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑆 ↾s 𝐴) = (𝑆 ↾s 𝐵))
32	31	adantr 483	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 ↾s 𝐴) = (𝑆 ↾s 𝐵))
33	7	adantl 484	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑆 ∈ 𝑉)
34	10	ressid 16561	. . . . . . . 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 2862	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = 𝑆)
37		df-base 16491	. . . . . . . 8 ⊢ Base = Slot 1
38		1nn 11651	. . . . . . . 8 ⊢ 1 ∈ ℕ
39		ressval3d.f	. . . . . . . 8 ⊢ (𝜑 → Fun 𝑆)
40		ressval3d.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ dom 𝑆)
41	18, 40	eqeltrrid 2920	. . . . . . . 8 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)
42	37, 38, 7, 39, 41	setsidvald 16516	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
43	42	adantl 484	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
44	18	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐸 = (Base‘ndx))
45		simpl 485	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐴 = 𝐵)
46	45, 10	syl6eq 2874	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐴 = (Base‘𝑆))
47	44, 46	opeq12d 4813	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (Base‘𝑆)⟩)
48	47	eqcomd 2829	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → ⟨(Base‘ndx), (Base‘𝑆)⟩ = ⟨𝐸, 𝐴⟩)
49	48	oveq2d 7174	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
50	36, 43, 49	3eqtrd 2862	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
51	50	ex 415	. . . 4 ⊢ (𝐴 = 𝐵 → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
52	29, 51	jaoi 853	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵) → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
53	5, 52	sylbi 219	. 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 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ∩ cin 3937   ⊆ wss 3938   ⊊ wpss 3939  ⟨cop 4575  dom cdm 5557  Fun wfun 6351  ‘cfv 6357  (class class class)co 7158  1c1 10540  ndxcnx 16482   sSet csts 16483  Basecbs 16485   ↾_s cress 16486

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-1cn 10597  ax-addcl 10599

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-nn 11641  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493

This theorem is referenced by:  estrres  17391