Theorem ressinbasd 12695

Description: Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.)

Hypotheses

Ref	Expression
ressidbasd.1	⊢ (𝜑 → 𝐵 = (Base‘𝑊))
ressidbasd.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
ressidbasd.w	⊢ (𝜑 → 𝑊 ∈ 𝑉)

Assertion

Ref	Expression
ressinbasd	⊢ (𝜑 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Proof of Theorem ressinbasd

Step	Hyp	Ref	Expression
1		ressidbasd.1	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑊))
2		inidm 3369	. . . . . . . 8 ⊢ (𝐵 ∩ 𝐵) = 𝐵
3	1	ineq2d 3361	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∩ 𝐵) = (𝐵 ∩ (Base‘𝑊)))
4	2, 3	eqtr3id 2240	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ (Base‘𝑊)))
5	1, 4	eqtr3d 2228	. . . . . 6 ⊢ (𝜑 → (Base‘𝑊) = (𝐵 ∩ (Base‘𝑊)))
6	5	ineq2d 3361	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))))
7		inass 3370	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
8	6, 7	eqtr4di 2244	. . . 4 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
9	8	opeq2d 3812	. . 3 ⊢ (𝜑 → ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩)
10	9	oveq2d 5935	. 2 ⊢ (𝜑 → (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
11		ressidbasd.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
12		ressidbasd.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
13		ressvalsets 12685	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
14	11, 12, 13	syl2anc 411	. 2 ⊢ (𝜑 → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
15		inex1g 4166	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
16	12, 15	syl 14	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
17		ressvalsets 12685	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
18	11, 16, 17	syl2anc 411	. 2 ⊢ (𝜑 → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
19	10, 14, 18	3eqtr4d 2236	1 ⊢ (𝜑 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∩ cin 3153  ⟨cop 3622  ‘cfv 5255  (class class class)co 5919  ndxcnx 12618   sSet csts 12619  Basecbs 12621   ↾_s cress 12622

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629

This theorem is referenced by: (None)