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Theorem dibfna 35461
 Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)
Hypotheses
Ref Expression
dibfna.h 𝐻 = (LHyp‘𝐾)
dibfna.j 𝐽 = ((DIsoA‘𝐾)‘𝑊)
dibfna.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibfna ((𝐾𝑉𝑊𝐻) → 𝐼 Fn dom 𝐽)

Proof of Theorem dibfna
Dummy variables 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6113 . . . 4 (𝐽𝑦) ∈ V
2 snex 4835 . . . 4 {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))} ∈ V
31, 2xpex 6860 . . 3 ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) ∈ V
4 eqid 2610 . . 3 (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) = (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}))
53, 4fnmpti 5935 . 2 (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽
6 eqid 2610 . . . 4 (Base‘𝐾) = (Base‘𝐾)
7 dibfna.h . . . 4 𝐻 = (LHyp‘𝐾)
8 eqid 2610 . . . 4 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9 eqid 2610 . . . 4 (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
10 dibfna.j . . . 4 𝐽 = ((DIsoA‘𝐾)‘𝑊)
11 dibfna.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
126, 7, 8, 9, 10, 11dibfval 35448 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})))
1312fneq1d 5895 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼 Fn dom 𝐽 ↔ (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽))
145, 13mpbiri 247 1 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn dom 𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125   ↦ cmpt 4643   I cid 4948   × cxp 5036  dom cdm 5038   ↾ cres 5040   Fn wfn 5799  ‘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:  dibdiadm  35462  dibfnN  35463  dibclN  35469
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