Theorem ressinbasd 13287

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 3430	. . . . . . . 8 ⊢ (𝐵 ∩ 𝐵) = 𝐵
3	1	ineq2d 3422	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∩ 𝐵) = (𝐵 ∩ (Base‘𝑊)))
4	2, 3	eqtr3id 2279	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ (Base‘𝑊)))
5	1, 4	eqtr3d 2267	. . . . . 6 ⊢ (𝜑 → (Base‘𝑊) = (𝐵 ∩ (Base‘𝑊)))
6	5	ineq2d 3422	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))))
7		inass 3431	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
8	6, 7	eqtr4di 2283	. . . 4 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
9	8	opeq2d 3890	. . 3 ⊢ (𝜑 → ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩)
10	9	oveq2d 6066	. 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 13277	. . 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 4246	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
16	12, 15	syl 14	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
17		ressvalsets 13277	. . 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 2275	1 ⊢ (𝜑 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2813   ∩ cin 3210  ⟨cop 3692  ‘cfv 5352  (class class class)co 6050  ndxcnx 13209   sSet csts 13210  Basecbs 13212   ↾_s cress 13213

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220

This theorem is referenced by: (None)