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Theorem dibfna 41137
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 6920 . . . 4 (𝐽𝑦) ∈ V
2 snex 5442 . . . 4 {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))} ∈ V
31, 2xpex 7772 . . 3 ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) ∈ V
4 eqid 2735 . . 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 6712 . 2 (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽
6 eqid 2735 . . . 4 (Base‘𝐾) = (Base‘𝐾)
7 dibfna.h . . . 4 𝐻 = (LHyp‘𝐾)
8 eqid 2735 . . . 4 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9 eqid 2735 . . . 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 41124 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})))
1312fneq1d 6662 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼 Fn dom 𝐽 ↔ (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽))
145, 13mpbiri 258 1 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn dom 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {csn 4631  cmpt 5231   I cid 5582   × cxp 5687  dom cdm 5689  cres 5691   Fn wfn 6558  cfv 6563  Basecbs 17245  LHypclh 39967  LTrncltrn 40084  DIsoAcdia 41011  DIsoBcdib 41121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-dib 41122
This theorem is referenced by:  dibdiadm  41138  dibfnN  41139  dibclN  41145
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