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Theorem dicfnN 38498
 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 5034 . . . . . . 7 (𝑝 = 𝑞 → (𝑝 𝑊𝑞 𝑊))
21notbid 321 . . . . . 6 (𝑝 = 𝑞 → (¬ 𝑝 𝑊 ↔ ¬ 𝑞 𝑊))
32elrab 3628 . . . . 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 2798 . . . . . . 7 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
8 eqid 2798 . . . . . . 7 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9 eqid 2798 . . . . . . 7 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
10 dicfn.i . . . . . . 7 𝐼 = ((DIsoC‘𝐾)‘𝑊)
114, 5, 6, 7, 8, 9, 10dicval 38491 . . . . . 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 6659 . . . . . 6 (𝐼𝑞) ∈ V
1311, 12eqeltrrdi 2899 . . . . 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 596 . . . 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 3149 . . 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 2798 . . . 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 6461 . . 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 38490 . . 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 6417 . 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 260 1 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   class class class wbr 5031  {copab 5093   ↦ cmpt 5111   Fn wfn 6320  ‘cfv 6325  ℩crio 7093  lecple 16567  occoc 16568  Atomscatm 36578  LHypclh 37299  LTrncltrn 37416  TEndoctendo 38067  DIsoCcdic 38487 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-dic 38488 This theorem is referenced by:  dicdmN  38499
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