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Theorem dicfnN 36992
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 4807 . . . . . . 7 (𝑝 = 𝑞 → (𝑝 𝑊𝑞 𝑊))
21notbid 307 . . . . . 6 (𝑝 = 𝑞 → (¬ 𝑝 𝑊 ↔ ¬ 𝑞 𝑊))
32elrab 3504 . . . . 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 2760 . . . . . . 7 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
8 eqid 2760 . . . . . . 7 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9 eqid 2760 . . . . . . 7 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
10 dicfn.i . . . . . . 7 𝐼 = ((DIsoC‘𝐾)‘𝑊)
114, 5, 6, 7, 8, 9, 10dicval 36985 . . . . . 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 6363 . . . . . 6 (𝐼𝑞) ∈ V
1311, 12syl6eqelr 2848 . . . . 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 493 . . . 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 3104 . . 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 2760 . . . 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 6181 . . 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 36984 . . 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 6142 . 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 247 1 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340   class class class wbr 4804  {copab 4864  cmpt 4881   Fn wfn 6044  cfv 6049  crio 6774  lecple 16170  occoc 16171  Atomscatm 35071  LHypclh 35791  LTrncltrn 35908  TEndoctendo 36560  DIsoCcdic 36981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-dic 36982
This theorem is referenced by:  dicdmN  36993
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