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Theorem dibdmN 35926
Description: Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-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
dibdmN ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
Distinct variable groups:   𝑥,   𝑥,𝐵   𝑥,𝐾   𝑥,𝑊
Allowed substitution hints:   𝐻(𝑥)   𝐼(𝑥)   𝑉(𝑥)

Proof of Theorem dibdmN
StepHypRef Expression
1 dibfn.b . . 3 𝐵 = (Base‘𝐾)
2 dibfn.l . . 3 = (le‘𝐾)
3 dibfn.h . . 3 𝐻 = (LHyp‘𝐾)
4 dibfn.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
51, 2, 3, 4dibfnN 35925 . 2 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑥𝐵𝑥 𝑊})
6 fndm 5948 . 2 (𝐼 Fn {𝑥𝐵𝑥 𝑊} → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
75, 6syl 17 1 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911   class class class wbr 4613  dom cdm 5074   Fn wfn 5842  cfv 5847  Basecbs 15781  lecple 15869  LHypclh 34750  DIsoBcdib 35907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-disoa 35798  df-dib 35908
This theorem is referenced by:  dibglbN  35935  dibintclN  35936  dihglblem3N  36064
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