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Theorem dibfna 41156
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 6919 . . . 4 (𝐽𝑦) ∈ V
2 snex 5436 . . . 4 {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))} ∈ V
31, 2xpex 7773 . . 3 ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) ∈ V
4 eqid 2737 . . 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 6711 . 2 (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽
6 eqid 2737 . . . 4 (Base‘𝐾) = (Base‘𝐾)
7 dibfna.h . . . 4 𝐻 = (LHyp‘𝐾)
8 eqid 2737 . . . 4 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9 eqid 2737 . . . 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 41143 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})))
1312fneq1d 6661 . 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 1540  wcel 2108  {csn 4626  cmpt 5225   I cid 5577   × cxp 5683  dom cdm 5685  cres 5687   Fn wfn 6556  cfv 6561  Basecbs 17247  LHypclh 39986  LTrncltrn 40103  DIsoAcdia 41030  DIsoBcdib 41140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-dib 41141
This theorem is referenced by:  dibdiadm  41157  dibfnN  41158  dibclN  41164
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