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Theorem ressval2 17187
Description: Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypotheses
Ref Expression
ressbas.r 𝑅 = (π‘Š β†Ύs 𝐴)
ressbas.b 𝐡 = (Baseβ€˜π‘Š)
Assertion
Ref Expression
ressval2 ((Β¬ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩))

Proof of Theorem ressval2
StepHypRef Expression
1 ressbas.r . . . 4 𝑅 = (π‘Š β†Ύs 𝐴)
2 ressbas.b . . . 4 𝐡 = (Baseβ€˜π‘Š)
31, 2ressval 17185 . . 3 ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)))
4 iffalse 4532 . . 3 (Β¬ 𝐡 βŠ† 𝐴 β†’ if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩)) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩))
53, 4sylan9eqr 2788 . 2 ((Β¬ 𝐡 βŠ† 𝐴 ∧ (π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ)) β†’ 𝑅 = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩))
653impb 1112 1 ((Β¬ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3942   βŠ† wss 3943  ifcif 4523  βŸ¨cop 4629  β€˜cfv 6537  (class class class)co 7405   sSet csts 17105  ndxcnx 17135  Basecbs 17153   β†Ύs cress 17182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-ress 17183
This theorem is referenced by:  ressbas  17188  ressbasOLD  17189  resseqnbas  17195  resslemOLD  17196  ressinbas  17199  ressval3d  17200  ressval3dOLD  17201  ressress  17202  rescabs  17791  rescabsOLD  17792  symgvalstruct  19316  symgvalstructOLD  19317  mgpress  20054  mgpressOLD  20055  resssra  33192
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