Theorem ressinbasd 13220

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 3418	. . . . . . . 8 ⊢ (𝐵 ∩ 𝐵) = 𝐵
3	1	ineq2d 3410	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∩ 𝐵) = (𝐵 ∩ (Base‘𝑊)))
4	2, 3	eqtr3id 2278	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ (Base‘𝑊)))
5	1, 4	eqtr3d 2266	. . . . . 6 ⊢ (𝜑 → (Base‘𝑊) = (𝐵 ∩ (Base‘𝑊)))
6	5	ineq2d 3410	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))))
7		inass 3419	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
8	6, 7	eqtr4di 2282	. . . 4 ⊢ (𝜑 → (𝐴 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
9	8	opeq2d 3874	. . 3 ⊢ (𝜑 → ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩)
10	9	oveq2d 6044	. 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 13210	. . 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 4230	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
16	12, 15	syl 14	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
17		ressvalsets 13210	. . 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 2274	1 ⊢ (𝜑 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ∩ cin 3200  ⟨cop 3676  ‘cfv 5333  (class class class)co 6028  ndxcnx 13142   sSet csts 13143  Basecbs 13145   ↾_s cress 13146

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153

This theorem is referenced by: (None)