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Theorem diafn 38242
Description: Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)
Hypotheses
Ref Expression
diafn.b 𝐵 = (Base‘𝐾)
diafn.l = (le‘𝐾)
diafn.h 𝐻 = (LHyp‘𝐾)
diafn.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
Assertion
Ref Expression
diafn ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑥𝐵𝑥 𝑊})
Distinct variable groups:   𝑥,   𝑥,𝐵   𝑥,𝐾   𝑥,𝑊
Allowed substitution hints:   𝐻(𝑥)   𝐼(𝑥)   𝑉(𝑥)

Proof of Theorem diafn
Dummy variables 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6672 . . . 4 ((LTrn‘𝐾)‘𝑊) ∈ V
21rabex 5222 . . 3 {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑦} ∈ V
3 eqid 2824 . . 3 (𝑦 ∈ {𝑥𝐵𝑥 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑦}) = (𝑦 ∈ {𝑥𝐵𝑥 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑦})
42, 3fnmpti 6480 . 2 (𝑦 ∈ {𝑥𝐵𝑥 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑦}) Fn {𝑥𝐵𝑥 𝑊}
5 diafn.b . . . 4 𝐵 = (Base‘𝐾)
6 diafn.l . . . 4 = (le‘𝐾)
7 diafn.h . . . 4 𝐻 = (LHyp‘𝐾)
8 eqid 2824 . . . 4 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9 eqid 2824 . . . 4 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
10 diafn.i . . . 4 𝐼 = ((DIsoA‘𝐾)‘𝑊)
115, 6, 7, 8, 9, 10diafval 38239 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑦 ∈ {𝑥𝐵𝑥 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑦}))
1211fneq1d 6435 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼 Fn {𝑥𝐵𝑥 𝑊} ↔ (𝑦 ∈ {𝑥𝐵𝑥 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) 𝑦}) Fn {𝑥𝐵𝑥 𝑊}))
134, 12mpbiri 261 1 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑥𝐵𝑥 𝑊})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {crab 3137   class class class wbr 5053  cmpt 5133   Fn wfn 6339  cfv 6344  Basecbs 16481  lecple 16570  LHypclh 37192  LTrncltrn 37309  trLctrl 37366  DIsoAcdia 38236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-disoa 38237
This theorem is referenced by:  diadm  38243  diaelrnN  38253  diaf11N  38257
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