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Theorem dibvalrel 35470
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 5150 . . 3 Rel ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})
2 dibcl.h . . . . . . . 8 𝐻 = (LHyp‘𝐾)
3 eqid 2610 . . . . . . . 8 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
4 dibcl.i . . . . . . . 8 𝐼 = ((DIsoB‘𝐾)‘𝑊)
52, 3, 4dibdiadm 35462 . . . . . . 7 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = dom ((DIsoA‘𝐾)‘𝑊))
65eleq2d 2673 . . . . . 6 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊)))
76biimpa 500 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊))
8 eqid 2610 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
9 eqid 2610 . . . . . 6 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
10 eqid 2610 . . . . . 6 ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = ( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
118, 2, 9, 10, 3, 4dibval 35449 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}))
127, 11syldan 486 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}))
1312releqd 5126 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (Rel (𝐼𝑋) ↔ Rel ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × {( ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})))
141, 13mpbiri 247 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → Rel (𝐼𝑋))
15 rel0 5166 . . . 4 Rel ∅
16 ndmfv 6128 . . . . 5 𝑋 ∈ dom 𝐼 → (𝐼𝑋) = ∅)
1716releqd 5126 . . . 4 𝑋 ∈ dom 𝐼 → (Rel (𝐼𝑋) ↔ Rel ∅))
1815, 17mpbiri 247 . . 3 𝑋 ∈ dom 𝐼 → Rel (𝐼𝑋))
1918adantl 481 . 2 (((𝐾𝑉𝑊𝐻) ∧ ¬ 𝑋 ∈ dom 𝐼) → Rel (𝐼𝑋))
2014, 19pm2.61dan 828 1 ((𝐾𝑉𝑊𝐻) → Rel (𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  c0 3874  {csn 4125  cmpt 4643   I cid 4948   × cxp 5036  dom cdm 5038  cres 5040  Rel wrel 5043  cfv 5804  Basecbs 15695  LHypclh 34288  LTrncltrn 34405  DIsoAcdia 35335  DIsoBcdib 35445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-dib 35446
This theorem is referenced by:  dibglbN  35473  dib2dim  35550  dih2dimbALTN  35552
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