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Theorem dicfnN 40786
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 5152 . . . . . . 7 (𝑝 = 𝑞 → (𝑝 𝑊𝑞 𝑊))
21notbid 317 . . . . . 6 (𝑝 = 𝑞 → (¬ 𝑝 𝑊 ↔ ¬ 𝑞 𝑊))
32elrab 3679 . . . . 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 2725 . . . . . . 7 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
8 eqid 2725 . . . . . . 7 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9 eqid 2725 . . . . . . 7 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
10 dicfn.i . . . . . . 7 𝐼 = ((DIsoC‘𝐾)‘𝑊)
114, 5, 6, 7, 8, 9, 10dicval 40779 . . . . . 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 6909 . . . . . 6 (𝐼𝑞) ∈ V
1311, 12eqeltrrdi 2834 . . . . 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 592 . . . 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 3135 . . 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 2725 . . . 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 6696 . . 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 40778 . . 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 6648 . 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 394   = wceq 1533  wcel 2098  wral 3050  {crab 3418  Vcvv 3461   class class class wbr 5149  {copab 5211  cmpt 5232   Fn wfn 6544  cfv 6549  crio 7374  lecple 17243  occoc 17244  Atomscatm 38865  LHypclh 39587  LTrncltrn 39704  TEndoctendo 40355  DIsoCcdic 40775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-dic 40776
This theorem is referenced by:  dicdmN  40787
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