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Theorem dibfna 37768
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 6510 . . . 4 (𝐽𝑦) ∈ V
2 snex 5185 . . . 4 {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))} ∈ V
31, 2xpex 7292 . . 3 ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) ∈ V
4 eqid 2773 . . 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 6319 . 2 (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽
6 eqid 2773 . . . 4 (Base‘𝐾) = (Base‘𝐾)
7 dibfna.h . . . 4 𝐻 = (LHyp‘𝐾)
8 eqid 2773 . . . 4 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9 eqid 2773 . . . 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 37755 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})))
1312fneq1d 6277 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼 Fn dom 𝐽 ↔ (𝑦 ∈ dom 𝐽 ↦ ((𝐽𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽))
145, 13mpbiri 250 1 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn dom 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wcel 2051  {csn 4436  cmpt 5005   I cid 5308   × cxp 5402  dom cdm 5404  cres 5406   Fn wfn 6181  cfv 6186  Basecbs 16338  LHypclh 36598  LTrncltrn 36715  DIsoAcdia 37642  DIsoBcdib 37752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-dib 37753
This theorem is referenced by:  dibdiadm  37769  dibfnN  37770  dibclN  37776
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