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Theorem ressbasd 12526
Description: Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
Hypotheses
Ref Expression
ressbasd.r (πœ‘ β†’ 𝑅 = (π‘Š β†Ύs 𝐴))
ressbasd.b (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))
ressbasd.w (πœ‘ β†’ π‘Š ∈ 𝑋)
ressbasd.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
Assertion
Ref Expression
ressbasd (πœ‘ β†’ (𝐴 ∩ 𝐡) = (Baseβ€˜π‘…))

Proof of Theorem ressbasd
StepHypRef Expression
1 ressbasd.w . . 3 (πœ‘ β†’ π‘Š ∈ 𝑋)
2 ressbasd.a . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝑉)
3 inex1g 4139 . . . 4 (𝐴 ∈ 𝑉 β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) ∈ V)
42, 3syl 14 . . 3 (πœ‘ β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) ∈ V)
5 baseslid 12518 . . . 4 (Base = Slot (Baseβ€˜ndx) ∧ (Baseβ€˜ndx) ∈ β„•)
65setsslid 12512 . . 3 ((π‘Š ∈ 𝑋 ∧ (𝐴 ∩ (Baseβ€˜π‘Š)) ∈ V) β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) = (Baseβ€˜(π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩)))
71, 4, 6syl2anc 411 . 2 (πœ‘ β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) = (Baseβ€˜(π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩)))
8 ressbasd.b . . 3 (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))
98ineq2d 3336 . 2 (πœ‘ β†’ (𝐴 ∩ 𝐡) = (𝐴 ∩ (Baseβ€˜π‘Š)))
10 ressbasd.r . . . 4 (πœ‘ β†’ 𝑅 = (π‘Š β†Ύs 𝐴))
11 ressvalsets 12523 . . . . 5 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
121, 2, 11syl2anc 411 . . . 4 (πœ‘ β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
1310, 12eqtrd 2210 . . 3 (πœ‘ β†’ 𝑅 = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
1413fveq2d 5519 . 2 (πœ‘ β†’ (Baseβ€˜π‘…) = (Baseβ€˜(π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩)))
157, 9, 143eqtr4d 2220 1 (πœ‘ β†’ (𝐴 ∩ 𝐡) = (Baseβ€˜π‘…))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2737   ∩ cin 3128  βŸ¨cop 3595  β€˜cfv 5216  (class class class)co 5874  ndxcnx 12458   sSet csts 12459  Basecbs 12461   β†Ύs cress 12462
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-inn 8919  df-ndx 12464  df-slot 12465  df-base 12467  df-sets 12468  df-iress 12469
This theorem is referenced by:  ressbas2d  12527  ressbasssd  12528  ressressg  12533  grpressid  12930  subrgpropd  13367
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