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Theorem ressval2 16541
Description: Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypotheses
Ref Expression
ressbas.r 𝑅 = (𝑊s 𝐴)
ressbas.b 𝐵 = (Base‘𝑊)
Assertion
Ref Expression
ressval2 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))

Proof of Theorem ressval2
StepHypRef Expression
1 ressbas.r . . . 4 𝑅 = (𝑊s 𝐴)
2 ressbas.b . . . 4 𝐵 = (Base‘𝑊)
31, 2ressval 16539 . . 3 ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
4 iffalse 4472 . . 3 𝐵𝐴 → if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
53, 4sylan9eqr 2875 . 2 ((¬ 𝐵𝐴 ∧ (𝑊𝑋𝐴𝑌)) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
653impb 1107 1 ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cin 3932  wss 3933  ifcif 4463  cop 4563  cfv 6348  (class class class)co 7145  ndxcnx 16468   sSet csts 16469  Basecbs 16471  s cress 16472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-ress 16479
This theorem is referenced by:  ressbas  16542  resslem  16545  ressinbas  16548  ressval3d  16549  ressress  16550  rescabs  17091  mgpress  19179
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