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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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