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Theorem ressinbasd 12552
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 (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))
ressidbasd.a (πœ‘ β†’ 𝐴 ∈ 𝑋)
ressidbasd.w (πœ‘ β†’ π‘Š ∈ 𝑉)
Assertion
Ref Expression
ressinbasd (πœ‘ β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))

Proof of Theorem ressinbasd
StepHypRef Expression
1 ressidbasd.1 . . . . . . 7 (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))
2 inidm 3359 . . . . . . . 8 (𝐡 ∩ 𝐡) = 𝐡
31ineq2d 3351 . . . . . . . 8 (πœ‘ β†’ (𝐡 ∩ 𝐡) = (𝐡 ∩ (Baseβ€˜π‘Š)))
42, 3eqtr3id 2236 . . . . . . 7 (πœ‘ β†’ 𝐡 = (𝐡 ∩ (Baseβ€˜π‘Š)))
51, 4eqtr3d 2224 . . . . . 6 (πœ‘ β†’ (Baseβ€˜π‘Š) = (𝐡 ∩ (Baseβ€˜π‘Š)))
65ineq2d 3351 . . . . 5 (πœ‘ β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) = (𝐴 ∩ (𝐡 ∩ (Baseβ€˜π‘Š))))
7 inass 3360 . . . . 5 ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š)) = (𝐴 ∩ (𝐡 ∩ (Baseβ€˜π‘Š)))
86, 7eqtr4di 2240 . . . 4 (πœ‘ β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) = ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š)))
98opeq2d 3800 . . 3 (πœ‘ β†’ ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩ = ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š))⟩)
109oveq2d 5907 . 2 (πœ‘ β†’ (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩) = (π‘Š sSet ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š))⟩))
11 ressidbasd.w . . 3 (πœ‘ β†’ π‘Š ∈ 𝑉)
12 ressidbasd.a . . 3 (πœ‘ β†’ 𝐴 ∈ 𝑋)
13 ressvalsets 12542 . . 3 ((π‘Š ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
1411, 12, 13syl2anc 411 . 2 (πœ‘ β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
15 inex1g 4154 . . . 4 (𝐴 ∈ 𝑋 β†’ (𝐴 ∩ 𝐡) ∈ V)
1612, 15syl 14 . . 3 (πœ‘ β†’ (𝐴 ∩ 𝐡) ∈ V)
17 ressvalsets 12542 . . 3 ((π‘Š ∈ 𝑉 ∧ (𝐴 ∩ 𝐡) ∈ V) β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = (π‘Š sSet ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š))⟩))
1811, 16, 17syl2anc 411 . 2 (πœ‘ β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = (π‘Š sSet ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š))⟩))
1910, 14, 183eqtr4d 2232 1 (πœ‘ β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   = wceq 1364   ∈ wcel 2160  Vcvv 2752   ∩ cin 3143  βŸ¨cop 3610  β€˜cfv 5231  (class class class)co 5891  ndxcnx 12477   sSet csts 12478  Basecbs 12480   β†Ύs cress 12481
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1re 7923  ax-addrcl 7926
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-inn 8938  df-ndx 12483  df-slot 12484  df-base 12486  df-sets 12487  df-iress 12488
This theorem is referenced by: (None)
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