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Theorem dicfnN 39613
Description: Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dicfn.l = (le‘𝐾)
dicfn.a 𝐴 = (Atoms‘𝐾)
dicfn.h 𝐻 = (LHyp‘𝐾)
dicfn.i 𝐼 = ((DIsoC‘𝐾)‘𝑊)
Assertion
Ref Expression
dicfnN ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
Distinct variable groups:   ,𝑝   𝐴,𝑝   𝐾,𝑝   𝑊,𝑝
Allowed substitution hints:   𝐻(𝑝)   𝐼(𝑝)   𝑉(𝑝)

Proof of Theorem dicfnN
Dummy variables 𝑞 𝑓 𝑠 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 5106 . . . . . . 7 (𝑝 = 𝑞 → (𝑝 𝑊𝑞 𝑊))
21notbid 317 . . . . . 6 (𝑝 = 𝑞 → (¬ 𝑝 𝑊 ↔ ¬ 𝑞 𝑊))
32elrab 3643 . . . . 5 (𝑞 ∈ {𝑝𝐴 ∣ ¬ 𝑝 𝑊} ↔ (𝑞𝐴 ∧ ¬ 𝑞 𝑊))
4 dicfn.l . . . . . . 7 = (le‘𝐾)
5 dicfn.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
6 dicfn.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
7 eqid 2736 . . . . . . 7 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
8 eqid 2736 . . . . . . 7 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9 eqid 2736 . . . . . . 7 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
10 dicfn.i . . . . . . 7 𝐼 = ((DIsoC‘𝐾)‘𝑊)
114, 5, 6, 7, 8, 9, 10dicval 39606 . . . . . 6 (((𝐾𝑉𝑊𝐻) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) → (𝐼𝑞) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
12 fvex 6852 . . . . . 6 (𝐼𝑞) ∈ V
1311, 12eqeltrrdi 2847 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))} ∈ V)
143, 13sylan2b 594 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ 𝑞 ∈ {𝑝𝐴 ∣ ¬ 𝑝 𝑊}) → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))} ∈ V)
1514ralrimiva 3141 . . 3 ((𝐾𝑉𝑊𝐻) → ∀𝑞 ∈ {𝑝𝐴 ∣ ¬ 𝑝 𝑊} {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))} ∈ V)
16 eqid 2736 . . . 4 (𝑞 ∈ {𝑝𝐴 ∣ ¬ 𝑝 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}) = (𝑞 ∈ {𝑝𝐴 ∣ ¬ 𝑝 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
1716fnmpt 6638 . . 3 (∀𝑞 ∈ {𝑝𝐴 ∣ ¬ 𝑝 𝑊} {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))} ∈ V → (𝑞 ∈ {𝑝𝐴 ∣ ¬ 𝑝 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}) Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
1815, 17syl 17 . 2 ((𝐾𝑉𝑊𝐻) → (𝑞 ∈ {𝑝𝐴 ∣ ¬ 𝑝 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}) Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
194, 5, 6, 7, 8, 9, 10dicfval 39605 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑞 ∈ {𝑝𝐴 ∣ ¬ 𝑝 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}))
2019fneq1d 6592 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼 Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊} ↔ (𝑞 ∈ {𝑝𝐴 ∣ ¬ 𝑝 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}) Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊}))
2118, 20mpbird 256 1 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3062  {crab 3405  Vcvv 3443   class class class wbr 5103  {copab 5165  cmpt 5186   Fn wfn 6488  cfv 6493  crio 7308  lecple 17132  occoc 17133  Atomscatm 37692  LHypclh 38414  LTrncltrn 38531  TEndoctendo 39182  DIsoCcdic 39602
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-dic 39603
This theorem is referenced by:  dicdmN  39614
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