Theorem ressinbasd 12825

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 3381	. . . . . . . 8 ⊢ (𝐵 ∩ 𝐵) = 𝐵
3	1	ineq2d 3373	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∩ 𝐵) = (𝐵 ∩ (Base‘𝑊)))
4	2, 3	eqtr3id 2251	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ (Base‘𝑊)))
5	1, 4	eqtr3d 2239	. . . . . 6 ⊢ (𝜑 → (Base‘𝑊) = (𝐵 ∩ (Base‘𝑊)))
6	5	ineq2d 3373	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))))
7		inass 3382	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
8	6, 7	eqtr4di 2255	. . . 4 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
9	8	opeq2d 3825	. . 3 ⊢ (𝜑 → ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩)
10	9	oveq2d 5950	. 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 12815	. . 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 4179	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
16	12, 15	syl 14	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
17		ressvalsets 12815	. . 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 2247	1 ⊢ (𝜑 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   = wceq 1372   ∈ wcel 2175  Vcvv 2771   ∩ cin 3164  ⟨cop 3635  ‘cfv 5268  (class class class)co 5934  ndxcnx 12748   sSet csts 12749  Basecbs 12751   ↾_s cress 12752

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1re 8001  ax-addrcl 8004

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-iota 5229  df-fun 5270  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-inn 9019  df-ndx 12754  df-slot 12755  df-base 12757  df-sets 12758  df-iress 12759

This theorem is referenced by: (None)