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Theorem dibfnN 39551
Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibfn.b 𝐵 = (Base‘𝐾)
dibfn.l = (le‘𝐾)
dibfn.h 𝐻 = (LHyp‘𝐾)
dibfn.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibfnN ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑥𝐵𝑥 𝑊})
Distinct variable groups:   𝑥,   𝑥,𝐵   𝑥,𝐾   𝑥,𝑊
Allowed substitution hints:   𝐻(𝑥)   𝐼(𝑥)   𝑉(𝑥)

Proof of Theorem dibfnN
StepHypRef Expression
1 dibfn.h . . 3 𝐻 = (LHyp‘𝐾)
2 eqid 2736 . . 3 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
3 dibfn.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
41, 2, 3dibfna 39549 . 2 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn dom ((DIsoA‘𝐾)‘𝑊))
5 dibfn.b . . . 4 𝐵 = (Base‘𝐾)
6 dibfn.l . . . 4 = (le‘𝐾)
75, 6, 1, 2diadm 39430 . . 3 ((𝐾𝑉𝑊𝐻) → dom ((DIsoA‘𝐾)‘𝑊) = {𝑥𝐵𝑥 𝑊})
87fneq2d 6593 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼 Fn dom ((DIsoA‘𝐾)‘𝑊) ↔ 𝐼 Fn {𝑥𝐵𝑥 𝑊}))
94, 8mpbid 231 1 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑥𝐵𝑥 𝑊})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3405   class class class wbr 5103  dom cdm 5631   Fn wfn 6488  cfv 6493  Basecbs 17037  lecple 17094  LHypclh 38379  DIsoAcdia 39423  DIsoBcdib 39533
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-disoa 39424  df-dib 39534
This theorem is referenced by:  dibdmN  39552  dibf11N  39556
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