Theorem ressval3d 16882

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 4030	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))
3		dfpss3 4017	. . . . 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 274	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵))
6		simplr 765	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ¬ 𝐵 ⊆ 𝐴)
7		ressval3d.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
8	7	adantl 481	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑆 ∈ 𝑉)
9		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ 𝐵)
10		ressval3d.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑆)
11	10	fvexi 6770	. . . . . . . . 9 ⊢ 𝐵 ∈ V
12	11	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
13		ssexg 5242	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
14	9, 12, 13	syl2an 595	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐴 ∈ V)
15		ressval3d.r	. . . . . . . 8 ⊢ 𝑅 = (𝑆 ↾s 𝐴)
16	15, 10	ressval2 16872	. . . . . . 7 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
17	6, 8, 14, 16	syl3anc 1369	. . . . . 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 3900	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
21	20	biimpi 215	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
22	21	eqcomd 2744	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (𝐴 ∩ 𝐵))
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐴 ∩ 𝐵))
24	23	adantr 480	. . . . . . . . 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 2744	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩ = ⟨𝐸, 𝐴⟩)
27	26	oveq2d 7271	. . . . . 6 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
28	17, 27	eqtrd 2778	. . . . 5 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
29	28	ex 412	. . . 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 7263	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑆 ↾s 𝐴) = (𝑆 ↾s 𝐵))
32	31	adantr 480	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 ↾s 𝐴) = (𝑆 ↾s 𝐵))
33	7	adantl 481	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑆 ∈ 𝑉)
34	10	ressid 16880	. . . . . . . 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 2782	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = 𝑆)
37		baseid 16843	. . . . . . . 8 ⊢ Base = Slot (Base‘ndx)
38		ressval3d.f	. . . . . . . 8 ⊢ (𝜑 → Fun 𝑆)
39		ressval3d.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ dom 𝑆)
40	18, 39	eqeltrrid 2844	. . . . . . . 8 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)
41	37, 7, 38, 40	setsidvald 16828	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
42	41	adantl 481	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
43	18	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐸 = (Base‘ndx))
44		simpl 482	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐴 = 𝐵)
45	44, 10	eqtrdi 2795	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐴 = (Base‘𝑆))
46	43, 45	opeq12d 4809	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (Base‘𝑆)⟩)
47	46	eqcomd 2744	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → ⟨(Base‘ndx), (Base‘𝑆)⟩ = ⟨𝐸, 𝐴⟩)
48	47	oveq2d 7271	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
49	36, 42, 48	3eqtrd 2782	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
50	49	ex 412	. . . 4 ⊢ (𝐴 = 𝐵 → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
51	29, 50	jaoi 853	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵) → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
52	5, 51	sylbi 216	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
53	1, 52	mpcom 38	1 ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883   ⊊ wpss 3884  ⟨cop 4564  dom cdm 5580  Fun wfun 6412  ‘cfv 6418  (class class class)co 7255   sSet csts 16792  ndxcnx 16822  Basecbs 16840   ↾_s cress 16867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-1cn 10860  ax-addcl 10862

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  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 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-nn 11904  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868

This theorem is referenced by:  estrres  17772