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Theorem dibvalrel 38179
Description: The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.)
Hypotheses
Ref Expression
dibcl.h 𝐻 = (LHyp‘𝐾)
dibcl.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibvalrel ((𝐾𝑉𝑊𝐻) → Rel (𝐼𝑋))

Proof of Theorem dibvalrel
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 relxp 5566 . . 3 Rel ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})
2 dibcl.h . . . . . . . 8 𝐻 = (LHyp‘𝐾)
3 eqid 2818 . . . . . . . 8 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
4 dibcl.i . . . . . . . 8 𝐼 = ((DIsoB‘𝐾)‘𝑊)
52, 3, 4dibdiadm 38171 . . . . . . 7 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = dom ((DIsoA‘𝐾)‘𝑊))
65eleq2d 2895 . . . . . 6 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊)))
76biimpa 477 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊))
8 eqid 2818 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
9 eqid 2818 . . . . . 6 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
10 eqid 2818 . . . . . 6 ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
118, 2, 9, 10, 3, 4dibval 38158 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}))
127, 11syldan 591 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}))
1312releqd 5646 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (Rel (𝐼𝑋) ↔ Rel ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})))
141, 13mpbiri 259 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → Rel (𝐼𝑋))
15 rel0 5665 . . . 4 Rel ∅
16 ndmfv 6693 . . . . 5 𝑋 ∈ dom 𝐼 → (𝐼𝑋) = ∅)
1716releqd 5646 . . . 4 𝑋 ∈ dom 𝐼 → (Rel (𝐼𝑋) ↔ Rel ∅))
1815, 17mpbiri 259 . . 3 𝑋 ∈ dom 𝐼 → Rel (𝐼𝑋))
1918adantl 482 . 2 (((𝐾𝑉𝑊𝐻) ∧ ¬ 𝑋 ∈ dom 𝐼) → Rel (𝐼𝑋))
2014, 19pm2.61dan 809 1 ((𝐾𝑉𝑊𝐻) → Rel (𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  c0 4288  {csn 4557  cmpt 5137   I cid 5452   × cxp 5546  dom cdm 5548  cres 5550  Rel wrel 5553  cfv 6348  Basecbs 16471  LHypclh 37000  LTrncltrn 37117  DIsoAcdia 38044  DIsoBcdib 38154
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-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
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-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-dib 38155
This theorem is referenced by:  dibglbN  38182  dib2dim  38259  dih2dimbALTN  38261
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