Theorem ressinbasd 13428

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 3440	. . . . . . . 8 ⊢ (𝐵 ∩ 𝐵) = 𝐵
3	1	ineq2d 3432	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∩ 𝐵) = (𝐵 ∩ (Base‘𝑊)))
4	2, 3	eqtr3id 2285	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ (Base‘𝑊)))
5	1, 4	eqtr3d 2273	. . . . . 6 ⊢ (𝜑 → (Base‘𝑊) = (𝐵 ∩ (Base‘𝑊)))
6	5	ineq2d 3432	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))))
7		inass 3441	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
8	6, 7	eqtr4di 2289	. . . 4 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
9	8	opeq2d 3911	. . 3 ⊢ (𝜑 → ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩)
10	9	oveq2d 6101	. 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 13418	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
14	11, 12, 13	syl2anc 415	. 2 ⊢ (𝜑 → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
15		inex1g 4269	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
16	12, 15	syl 14	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
17		ressvalsets 13418	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
18	11, 16, 17	syl2anc 415	. 2 ⊢ (𝜑 → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
19	10, 14, 18	3eqtr4d 2281	1 ⊢ (𝜑 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∩ cin 3219  ⟨cop 3712  ‘cfv 5377  (class class class)co 6085  ndxcnx 13349   sSet csts 13350  Basecbs 13352   ↾_s cress 13353

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-inn 9305  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360

This theorem is used by: (None)