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Theorem dicfnN 38186
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 5066 . . . . . . 7 (𝑝 = 𝑞 → (𝑝 𝑊𝑞 𝑊))
21notbid 319 . . . . . 6 (𝑝 = 𝑞 → (¬ 𝑝 𝑊 ↔ ¬ 𝑞 𝑊))
32elrab 3684 . . . . 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 2826 . . . . . . 7 ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
8 eqid 2826 . . . . . . 7 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9 eqid 2826 . . . . . . 7 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
10 dicfn.i . . . . . . 7 𝐼 = ((DIsoC‘𝐾)‘𝑊)
114, 5, 6, 7, 8, 9, 10dicval 38179 . . . . . 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 6680 . . . . . 6 (𝐼𝑞) ∈ V
1311, 12syl6eqelr 2927 . . . . 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 593 . . . 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 3187 . . 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 2826 . . . 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 6485 . . 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 38178 . . 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 6443 . 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 258 1 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  {crab 3147  Vcvv 3500   class class class wbr 5063  {copab 5125  cmpt 5143   Fn wfn 6347  cfv 6352  crio 7105  lecple 16562  occoc 16563  Atomscatm 36266  LHypclh 36987  LTrncltrn 37104  TEndoctendo 37755  DIsoCcdic 38175
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-dic 38176
This theorem is referenced by:  dicdmN  38187
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