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Theorem dibvalrel 40034
Description: The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.)
Hypotheses
Ref Expression
dibcl.h 𝐻 = (LHypβ€˜πΎ)
dibcl.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dibvalrel ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ Rel (πΌβ€˜π‘‹))

Proof of Theorem dibvalrel
Dummy variable β„Ž is distinct from all other variables.
StepHypRef Expression
1 relxp 5695 . . 3 Rel ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘‹) Γ— {(β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))})
2 dibcl.h . . . . . . . 8 𝐻 = (LHypβ€˜πΎ)
3 eqid 2733 . . . . . . . 8 ((DIsoAβ€˜πΎ)β€˜π‘Š) = ((DIsoAβ€˜πΎ)β€˜π‘Š)
4 dibcl.i . . . . . . . 8 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
52, 3, 4dibdiadm 40026 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ dom 𝐼 = dom ((DIsoAβ€˜πΎ)β€˜π‘Š))
65eleq2d 2820 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘Š)))
76biimpa 478 . . . . 5 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ 𝑋 ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘Š))
8 eqid 2733 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
9 eqid 2733 . . . . . 6 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
10 eqid 2733 . . . . . 6 (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ))) = (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))
118, 2, 9, 10, 3, 4dibval 40013 . . . . 5 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘Š)) β†’ (πΌβ€˜π‘‹) = ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘‹) Γ— {(β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))}))
127, 11syldan 592 . . . 4 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ (πΌβ€˜π‘‹) = ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘‹) Γ— {(β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))}))
1312releqd 5779 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ (Rel (πΌβ€˜π‘‹) ↔ Rel ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘‹) Γ— {(β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ↦ ( I β†Ύ (Baseβ€˜πΎ)))})))
141, 13mpbiri 258 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ Rel (πΌβ€˜π‘‹))
15 rel0 5800 . . . 4 Rel βˆ…
16 ndmfv 6927 . . . . 5 (Β¬ 𝑋 ∈ dom 𝐼 β†’ (πΌβ€˜π‘‹) = βˆ…)
1716releqd 5779 . . . 4 (Β¬ 𝑋 ∈ dom 𝐼 β†’ (Rel (πΌβ€˜π‘‹) ↔ Rel βˆ…))
1815, 17mpbiri 258 . . 3 (Β¬ 𝑋 ∈ dom 𝐼 β†’ Rel (πΌβ€˜π‘‹))
1918adantl 483 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ Β¬ 𝑋 ∈ dom 𝐼) β†’ Rel (πΌβ€˜π‘‹))
2014, 19pm2.61dan 812 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ Rel (πΌβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ…c0 4323  {csn 4629   ↦ cmpt 5232   I cid 5574   Γ— cxp 5675  dom cdm 5677   β†Ύ cres 5679  Rel wrel 5682  β€˜cfv 6544  Basecbs 17144  LHypclh 38855  LTrncltrn 38972  DIsoAcdia 39899  DIsoBcdib 40009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-dib 40010
This theorem is referenced by:  dibglbN  40037  dib2dim  40114  dih2dimbALTN  40116
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