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Theorem dibelval3 37035
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 2764 . . . 4 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7 dibval3.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dibval2 37032 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
98eleq2d 2829 . 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 36921 . . . . . 6 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (𝑓𝑇 ∧ (𝑅𝑓) 𝑋)))
1211anbi1d 623 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩)))
13 an13 637 . . . . . . . 8 ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑠 ∈ { 0 } ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
14 velsn 4349 . . . . . . . . 9 (𝑠 ∈ { 0 } ↔ 𝑠 = 0 )
1514anbi1i 617 . . . . . . . 8 ((𝑠 ∈ { 0 } ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)) ↔ (𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
1613, 15bitri 266 . . . . . . 7 ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
1716exbii 1943 . . . . . 6 (∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ ∃𝑠(𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
184fvexi 6388 . . . . . . . . 9 𝑇 ∈ V
1918mptex 6678 . . . . . . . 8 (𝑔𝑇 ↦ ( I ↾ 𝐵)) ∈ V
205, 19eqeltri 2839 . . . . . . 7 0 ∈ V
21 opeq2 4559 . . . . . . . . 9 (𝑠 = 0 → ⟨𝑓, 𝑠⟩ = ⟨𝑓, 0 ⟩)
2221eqeq2d 2774 . . . . . . . 8 (𝑠 = 0 → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨𝑓, 0 ⟩))
2322anbi2d 622 . . . . . . 7 (𝑠 = 0 → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩)))
2420, 23ceqsexv 3394 . . . . . 6 (∃𝑠(𝑠 = 0 ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
2517, 24bitri 266 . . . . 5 (∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
26 anass 460 . . . . . 6 (((𝑓𝑇𝑌 = ⟨𝑓, 0 ⟩) ∧ (𝑅𝑓) 𝑋) ↔ (𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)))
27 an32 636 . . . . . 6 (((𝑓𝑇𝑌 = ⟨𝑓, 0 ⟩) ∧ (𝑅𝑓) 𝑋) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
2826, 27bitr3i 268 . . . . 5 ((𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩))
2912, 25, 283bitr4g 305 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋))))
3029exbidv 2016 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ ∃𝑓(𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋))))
31 elxp 5299 . . 3 (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) ↔ ∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })))
32 df-rex 3060 . . 3 (∃𝑓𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋) ↔ ∃𝑓(𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)))
3330, 31, 323bitr4g 305 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) ↔ ∃𝑓𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)))
349, 33bitrd 270 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ (𝐼𝑋) ↔ ∃𝑓𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wrex 3055  Vcvv 3349  {csn 4333  cop 4339   class class class wbr 4808  cmpt 4887   I cid 5183   × cxp 5274  cres 5278  cfv 6067  Basecbs 16131  lecple 16222  LHypclh 35872  LTrncltrn 35989  trLctrl 36046  DIsoAcdia 36916  DIsoBcdib 37026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-disoa 36917  df-dib 37027
This theorem is referenced by:  cdlemn11pre  37098  dihord2pre  37113
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