Theorem ressval3d 13100

Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)

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.r	. . 3 ⊢ 𝑅 = (𝑆 ↾s 𝐴)
2		ressval3d.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
3		ressval3d.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
4		basfn 13086	. . . . . . 7 ⊢ Base Fn V
5	2	elexd 2813	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V)
6		funfvex 5643	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
7	6	funfni 5422	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
8	4, 5, 7	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
9	3, 8	eqeltrid 2316	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
10		ressval3d.u	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
11	9, 10	ssexd 4223	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12		ressvalsets 13092	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝑆 ↾s 𝐴) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
13	2, 11, 12	syl2anc 411	. . 3 ⊢ (𝜑 → (𝑆 ↾s 𝐴) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
14	1, 13	eqtrid 2274	. 2 ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
15		ressval3d.e	. . . . 5 ⊢ 𝐸 = (Base‘ndx)
16	15	a1i 9	. . . 4 ⊢ (𝜑 → 𝐸 = (Base‘ndx))
17		df-ss 3210	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
18	10, 17	sylib 122	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐴)
19	3	ineq2i 3402	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ (Base‘𝑆))
20	18, 19	eqtr3di 2277	. . . 4 ⊢ (𝜑 → 𝐴 = (𝐴 ∩ (Base‘𝑆)))
21	16, 20	opeq12d 3864	. . 3 ⊢ (𝜑 → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩)
22	21	oveq2d 6016	. 2 ⊢ (𝜑 → (𝑆 sSet ⟨𝐸, 𝐴⟩) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
23	14, 22	eqtr4d 2265	1 ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∩ cin 3196   ⊆ wss 3197  ⟨cop 3669  dom cdm 4718  Fun wfun 5311   Fn wfn 5312  ‘cfv 5317  (class class class)co 6000  ndxcnx 13024   sSet csts 13025  Basecbs 13027   ↾_s cress 13028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-iress 13035

This theorem is referenced by: (None)