Theorem ressval3d 12750

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 12736	. . . . . . 7 ⊢ Base Fn V
5	2	elexd 2776	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V)
6		funfvex 5575	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
7	6	funfni 5358	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
8	4, 5, 7	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
9	3, 8	eqeltrid 2283	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
10		ressval3d.u	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
11	9, 10	ssexd 4173	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12		ressvalsets 12742	. . . 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 2241	. 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 3170	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
18	10, 17	sylib 122	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐴)
19	3	ineq2i 3361	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ (Base‘𝑆))
20	18, 19	eqtr3di 2244	. . . 4 ⊢ (𝜑 → 𝐴 = (𝐴 ∩ (Base‘𝑆)))
21	16, 20	opeq12d 3816	. . 3 ⊢ (𝜑 → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩)
22	21	oveq2d 5938	. 2 ⊢ (𝜑 → (𝑆 sSet ⟨𝐸, 𝐴⟩) = (𝑆 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑆))⟩))
23	14, 22	eqtr4d 2232	1 ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∩ cin 3156   ⊆ wss 3157  ⟨cop 3625  dom cdm 4663  Fun wfun 5252   Fn wfn 5253  ‘cfv 5258  (class class class)co 5922  ndxcnx 12675   sSet csts 12676  Basecbs 12678   ↾_s cress 12679

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686

This theorem is referenced by: (None)