Theorem ressval3dOLD 16858

Description: Obsolete version of ressval3d 16857 as of 17-Oct-2024. Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.) (New usage is discouraged.) (Proof modification is discouraged.)

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
ressval3dOLD	⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Proof of Theorem ressval3dOLD

Step	Hyp	Ref	Expression
1		ressval3d.u	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sspss 4031	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
3		dfpss3 4018	. . . . 5 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴))
4	3	orbi1i 914	. . . 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 769	. . . . . . 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 6767	. . . . . . . . 9 ⊢ 𝐵 ∈ V
12	11	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
13		ssexg 5240	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
14	9, 12, 13	syl2an 599	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐴 ∈ V)
15		ressval3d.r	. . . . . . . 8 ⊢ 𝑅 = (𝑆 ↾s 𝐴)
16	15, 10	ressval2 16847	. . . . . . 7 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
17	6, 8, 14, 16	syl3anc 1373	. . . . . 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 3901	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
21	20	biimpi 219	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
22	21	eqcomd 2745	. . . . . . . . . . 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 4809	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)
26	25	eqcomd 2745	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩ = ⟨𝐸, 𝐴⟩)
27	26	oveq2d 7268	. . . . . 6 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
28	17, 27	eqtrd 2779	. . . . 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 7260	. . . . . . . 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 16855	. . . . . . . 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 2783	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = 𝑆)
37		df-base 16816	. . . . . . . 8 ⊢ Base = Slot 1
38		1nn 11889	. . . . . . . 8 ⊢ 1 ∈ ℕ
39		ressval3d.f	. . . . . . . 8 ⊢ (𝜑 → Fun 𝑆)
40		ressval3d.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ dom 𝑆)
41	18, 40	eqeltrrid 2845	. . . . . . . 8 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)
42	37, 38, 7, 39, 41	setsidvaldOLD 16804	. . . . . . 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 2796	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐴 = (Base‘𝑆))
47	44, 46	opeq12d 4809	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (Base‘𝑆)⟩)
48	47	eqcomd 2745	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → ⟨(Base‘ndx), (Base‘𝑆)⟩ = ⟨𝐸, 𝐴⟩)
49	48	oveq2d 7268	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
50	36, 43, 49	3eqtrd 2783	. . . . 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 857	. . 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 847   = wceq 1543   ∈ wcel 2112  Vcvv 3423   ∩ cin 3883   ⊆ wss 3884   ⊊ wpss 3885  ⟨cop 4564  dom cdm 5579  Fun wfun 6409  ‘cfv 6415  (class class class)co 7252  1c1 10778   sSet csts 16767  ndxcnx 16797  Basecbs 16815   ↾_s cress 16842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5216  ax-nul 5223  ax-pow 5282  ax-pr 5346  ax-un 7563  ax-cnex 10833  ax-1cn 10835  ax-addcl 10837

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3713  df-csb 3830  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5153  df-tr 5186  df-id 5479  df-eprel 5485  df-po 5493  df-so 5494  df-fr 5534  df-we 5536  df-xp 5585  df-rel 5586  df-cnv 5587  df-co 5588  df-dm 5589  df-rn 5590  df-res 5591  df-ima 5592  df-pred 6189  df-ord 6251  df-on 6252  df-lim 6253  df-suc 6254  df-iota 6373  df-fun 6417  df-fn 6418  df-f 6419  df-f1 6420  df-fo 6421  df-f1o 6422  df-fv 6423  df-ov 7255  df-oprab 7256  df-mpo 7257  df-om 7685  df-wrecs 8089  df-recs 8150  df-rdg 8188  df-nn 11879  df-sets 16768  df-slot 16786  df-ndx 16798  df-base 16816  df-ress 16843

This theorem is referenced by: (None)