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Theorem ressinbasd 12547
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 3356 . . . . . . . 8 (𝐡 ∩ 𝐡) = 𝐡
31ineq2d 3348 . . . . . . . 8 (πœ‘ β†’ (𝐡 ∩ 𝐡) = (𝐡 ∩ (Baseβ€˜π‘Š)))
42, 3eqtr3id 2234 . . . . . . 7 (πœ‘ β†’ 𝐡 = (𝐡 ∩ (Baseβ€˜π‘Š)))
51, 4eqtr3d 2222 . . . . . 6 (πœ‘ β†’ (Baseβ€˜π‘Š) = (𝐡 ∩ (Baseβ€˜π‘Š)))
65ineq2d 3348 . . . . 5 (πœ‘ β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) = (𝐴 ∩ (𝐡 ∩ (Baseβ€˜π‘Š))))
7 inass 3357 . . . . 5 ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š)) = (𝐴 ∩ (𝐡 ∩ (Baseβ€˜π‘Š)))
86, 7eqtr4di 2238 . . . 4 (πœ‘ β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) = ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š)))
98opeq2d 3797 . . 3 (πœ‘ β†’ ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩ = ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š))⟩)
109oveq2d 5904 . 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 12537 . . 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 4151 . . . 4 (𝐴 ∈ 𝑋 β†’ (𝐴 ∩ 𝐡) ∈ V)
1612, 15syl 14 . . 3 (πœ‘ β†’ (𝐴 ∩ 𝐡) ∈ V)
17 ressvalsets 12537 . . 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 2230 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 1363   ∈ wcel 2158  Vcvv 2749   ∩ cin 3140  βŸ¨cop 3607  β€˜cfv 5228  (class class class)co 5888  ndxcnx 12472   sSet csts 12473  Basecbs 12475   β†Ύs cress 12476
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-inn 8933  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483
This theorem is referenced by: (None)
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