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Theorem dibfnN 36762
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 2651 . . 3 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
3 dibfn.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
41, 2, 3dibfna 36760 . 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 36641 . . 3 ((𝐾𝑉𝑊𝐻) → dom ((DIsoA‘𝐾)‘𝑊) = {𝑥𝐵𝑥 𝑊})
87fneq2d 6020 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼 Fn dom ((DIsoA‘𝐾)‘𝑊) ↔ 𝐼 Fn {𝑥𝐵𝑥 𝑊}))
94, 8mpbid 222 1 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑥𝐵𝑥 𝑊})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {crab 2945   class class class wbr 4685  dom cdm 5143   Fn wfn 5921  cfv 5926  Basecbs 15904  lecple 15995  LHypclh 35588  DIsoAcdia 36634  DIsoBcdib 36744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-disoa 36635  df-dib 36745
This theorem is referenced by:  dibdmN  36763  dibf11N  36767
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