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Theorem dibfna 40866
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 6906 . . . 4 (𝐽𝑦) ∈ V
2 snex 5429 . . . 4 {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))} ∈ V
31, 2xpex 7753 . . 3 ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) ∈ V
4 eqid 2726 . . 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 6696 . 2 (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽
6 eqid 2726 . . . 4 (Base‘𝐾) = (Base‘𝐾)
7 dibfna.h . . . 4 𝐻 = (LHyp‘𝐾)
8 eqid 2726 . . . 4 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9 eqid 2726 . . . 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 40853 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})))
1312fneq1d 6645 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼 Fn dom 𝐽 ↔ (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽))
145, 13mpbiri 257 1 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn dom 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  {csn 4623  cmpt 5228   I cid 5571   × cxp 5672  dom cdm 5674  cres 5676   Fn wfn 6541  cfv 6546  Basecbs 17208  LHypclh 39696  LTrncltrn 39813  DIsoAcdia 40740  DIsoBcdib 40850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-dib 40851
This theorem is referenced by:  dibdiadm  40867  dibfnN  40868  dibclN  40874
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