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Theorem dibelval3 36753
 Description: Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.)
Hypotheses
Ref Expression
dibval3.b 𝐵 = (Base‘𝐾)
dibval3.l = (le‘𝐾)
dibval3.h 𝐻 = (LHyp‘𝐾)
dibval3.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibval3.r 𝑅 = ((trL‘𝐾)‘𝑊)
dibval3.o 0 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
dibval3.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibelval3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ (𝐼𝑋) ↔ ∃𝑓𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)))
Distinct variable groups:   𝑓,𝐾   𝑔,𝐾   𝑇,𝑓   𝑓,𝑊   𝑔,𝑊   𝑓,𝑋   ,𝑓   𝐵,𝑓   𝑓,𝐻   0 ,𝑓   𝑇,𝑔   𝑓,𝑉   𝑓,𝑌
Allowed substitution hints:   𝐵(𝑔)   𝑅(𝑓,𝑔)   𝐻(𝑔)   𝐼(𝑓,𝑔)   (𝑔)   𝑉(𝑔)   𝑋(𝑔)   𝑌(𝑔)   0 (𝑔)

Proof of Theorem dibelval3
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 dibval3.b . . . 4 𝐵 = (Base‘𝐾)
2 dibval3.l . . . 4 = (le‘𝐾)
3 dibval3.h . . . 4 𝐻 = (LHyp‘𝐾)
4 dibval3.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
5 dibval3.o . . . 4 0 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
6 eqid 2651 . . . 4 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7 dibval3.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dibval2 36750 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
98eleq2d 2716 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ (𝐼𝑋) ↔ 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })))
10 dibval3.r . . . . . . 7 𝑅 = ((trL‘𝐾)‘𝑊)
111, 2, 3, 4, 10, 6diaelval 36639 . . . . . 6 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (𝑓𝑇 ∧ (𝑅𝑓) 𝑋)))
1211anbi1d 741 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩)))
13 an13 857 . . . . . . . 8 ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑠 ∈ { 0 } ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
14 velsn 4226 . . . . . . . . 9 (𝑠 ∈ { 0 } ↔ 𝑠 = 0 )
1514anbi1i 731 . . . . . . . 8 ((𝑠 ∈ { 0 } ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)) ↔ (𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
1613, 15bitri 264 . . . . . . 7 ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
1716exbii 1814 . . . . . 6 (∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ ∃𝑠(𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
18 fvex 6239 . . . . . . . . . 10 ((LTrn‘𝐾)‘𝑊) ∈ V
194, 18eqeltri 2726 . . . . . . . . 9 𝑇 ∈ V
2019mptex 6527 . . . . . . . 8 (𝑔𝑇 ↦ ( I ↾ 𝐵)) ∈ V
215, 20eqeltri 2726 . . . . . . 7 0 ∈ V
22 opeq2 4434 . . . . . . . . 9 (𝑠 = 0 → ⟨𝑓, 𝑠⟩ = ⟨𝑓, 0 ⟩)
2322eqeq2d 2661 . . . . . . . 8 (𝑠 = 0 → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨𝑓, 0 ⟩))
2423anbi2d 740 . . . . . . 7 (𝑠 = 0 → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩)))
2521, 24ceqsexv 3273 . . . . . 6 (∃𝑠(𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
2617, 25bitri 264 . . . . 5 (∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
27 anass 682 . . . . . 6 (((𝑓𝑇𝑌 = ⟨𝑓, 0 ⟩) ∧ (𝑅𝑓) 𝑋) ↔ (𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)))
28 an32 856 . . . . . 6 (((𝑓𝑇𝑌 = ⟨𝑓, 0 ⟩) ∧ (𝑅𝑓) 𝑋) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
2927, 28bitr3i 266 . . . . 5 ((𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
3012, 26, 293bitr4g 303 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋))))
3130exbidv 1890 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ ∃𝑓(𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋))))
32 elxp 5165 . . 3 (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) ↔ ∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })))
33 df-rex 2947 . . 3 (∃𝑓𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋) ↔ ∃𝑓(𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)))
3431, 32, 333bitr4g 303 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) ↔ ∃𝑓𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)))
359, 34bitrd 268 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ (𝐼𝑋) ↔ ∃𝑓𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃wrex 2942  Vcvv 3231  {csn 4210  ⟨cop 4216   class class class wbr 4685   ↦ cmpt 4762   I cid 5052   × cxp 5141   ↾ cres 5145  ‘cfv 5926  Basecbs 15904  lecple 15995  LHypclh 35588  LTrncltrn 35705  trLctrl 35763  DIsoAcdia 36634  DIsoBcdib 36744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-disoa 36635  df-dib 36745 This theorem is referenced by:  cdlemn11pre  36816  dihord2pre  36831
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