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Theorem dicvalrelN 37333
 Description: The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dicvalrel.h 𝐻 = (LHyp‘𝐾)
dicvalrel.i 𝐼 = ((DIsoC‘𝐾)‘𝑊)
Assertion
Ref Expression
dicvalrelN ((𝐾𝑉𝑊𝐻) → Rel (𝐼𝑋))

Proof of Theorem dicvalrelN
Dummy variables 𝑓 𝑔 𝑝 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 5493 . . . 4 Rel {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}
2 eqid 2777 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
3 eqid 2777 . . . . . . . . . 10 (Atoms‘𝐾) = (Atoms‘𝐾)
4 dicvalrel.h . . . . . . . . . 10 𝐻 = (LHyp‘𝐾)
5 dicvalrel.i . . . . . . . . . 10 𝐼 = ((DIsoC‘𝐾)‘𝑊)
62, 3, 4, 5dicdmN 37332 . . . . . . . . 9 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊})
76eleq2d 2844 . . . . . . . 8 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼𝑋 ∈ {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊}))
8 breq1 4889 . . . . . . . . . 10 (𝑝 = 𝑋 → (𝑝(le‘𝐾)𝑊𝑋(le‘𝐾)𝑊))
98notbid 310 . . . . . . . . 9 (𝑝 = 𝑋 → (¬ 𝑝(le‘𝐾)𝑊 ↔ ¬ 𝑋(le‘𝐾)𝑊))
109elrab 3571 . . . . . . . 8 (𝑋 ∈ {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊} ↔ (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊))
117, 10syl6bb 279 . . . . . . 7 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊)))
1211biimpa 470 . . . . . 6 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊))
13 eqid 2777 . . . . . . 7 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
14 eqid 2777 . . . . . . 7 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
15 eqid 2777 . . . . . . 7 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
162, 3, 4, 13, 14, 15, 5dicval 37324 . . . . . 6 (((𝐾𝑉𝑊𝐻) ∧ (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊)) → (𝐼𝑋) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
1712, 16syldan 585 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
1817releqd 5451 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (Rel (𝐼𝑋) ↔ Rel {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}))
191, 18mpbiri 250 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → Rel (𝐼𝑋))
2019ex 403 . 2 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 → Rel (𝐼𝑋)))
21 rel0 5470 . . 3 Rel ∅
22 ndmfv 6476 . . . 4 𝑋 ∈ dom 𝐼 → (𝐼𝑋) = ∅)
2322releqd 5451 . . 3 𝑋 ∈ dom 𝐼 → (Rel (𝐼𝑋) ↔ Rel ∅))
2421, 23mpbiri 250 . 2 𝑋 ∈ dom 𝐼 → Rel (𝐼𝑋))
2520, 24pm2.61d1 173 1 ((𝐾𝑉𝑊𝐻) → Rel (𝐼𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106  {crab 3093  ∅c0 4140   class class class wbr 4886  {copab 4948  dom cdm 5355  Rel wrel 5360  ‘cfv 6135  ℩crio 6882  lecple 16345  occoc 16346  Atomscatm 35411  LHypclh 36132  LTrncltrn 36249  TEndoctendo 36900  DIsoCcdic 37320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-dic 37321 This theorem is referenced by: (None)
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