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Theorem dicvalrelN 38439
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 5673 . . . 4 Rel {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}
2 eqid 2822 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
3 eqid 2822 . . . . . . . . . 10 (Atoms‘𝐾) = (Atoms‘𝐾)
4 dicvalrel.h . . . . . . . . . 10 𝐻 = (LHyp‘𝐾)
5 dicvalrel.i . . . . . . . . . 10 𝐼 = ((DIsoC‘𝐾)‘𝑊)
62, 3, 4, 5dicdmN 38438 . . . . . . . . 9 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊})
76eleq2d 2899 . . . . . . . 8 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼𝑋 ∈ {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊}))
8 breq1 5045 . . . . . . . . . 10 (𝑝 = 𝑋 → (𝑝(le‘𝐾)𝑊𝑋(le‘𝐾)𝑊))
98notbid 321 . . . . . . . . 9 (𝑝 = 𝑋 → (¬ 𝑝(le‘𝐾)𝑊 ↔ ¬ 𝑋(le‘𝐾)𝑊))
109elrab 3655 . . . . . . . 8 (𝑋 ∈ {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊} ↔ (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊))
117, 10syl6bb 290 . . . . . . 7 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊)))
1211biimpa 480 . . . . . 6 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊))
13 eqid 2822 . . . . . . 7 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
14 eqid 2822 . . . . . . 7 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
15 eqid 2822 . . . . . . 7 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
162, 3, 4, 13, 14, 15, 5dicval 38430 . . . . . 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 594 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
1817releqd 5630 . . . 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 261 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → Rel (𝐼𝑋))
2019ex 416 . 2 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 → Rel (𝐼𝑋)))
21 rel0 5649 . . 3 Rel ∅
22 ndmfv 6682 . . . 4 𝑋 ∈ dom 𝐼 → (𝐼𝑋) = ∅)
2322releqd 5630 . . 3 𝑋 ∈ dom 𝐼 → (Rel (𝐼𝑋) ↔ Rel ∅))
2421, 23mpbiri 261 . 2 𝑋 ∈ dom 𝐼 → Rel (𝐼𝑋))
2520, 24pm2.61d1 183 1 ((𝐾𝑉𝑊𝐻) → Rel (𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2114  {crab 3134  c0 4265   class class class wbr 5042  {copab 5104  dom cdm 5532  Rel wrel 5537  cfv 6334  crio 7097  lecple 16563  occoc 16564  Atomscatm 36517  LHypclh 37238  LTrncltrn 37355  TEndoctendo 38006  DIsoCcdic 38426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-dic 38427
This theorem is referenced by: (None)
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