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Theorem dibelval3 35943
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 2621 . . . 4 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7 dibval3.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dibval2 35940 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
98eleq2d 2684 . 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 35829 . . . . . 6 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (𝑓𝑇 ∧ (𝑅𝑓) 𝑋)))
1211anbi1d 740 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩)))
13 an13 839 . . . . . . . 8 ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ (𝑠 ∈ { 0 } ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩)))
14 velsn 4169 . . . . . . . . 9 (𝑠 ∈ { 0 } ↔ 𝑠 = 0 )
1514anbi1i 730 . . . . . . . 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 1771 . . . . . 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 6163 . . . . . . . . . 10 ((LTrn‘𝐾)‘𝑊) ∈ V
194, 18eqeltri 2694 . . . . . . . . 9 𝑇 ∈ V
2019mptex 6446 . . . . . . . 8 (𝑔𝑇 ↦ ( I ↾ 𝐵)) ∈ V
215, 20eqeltri 2694 . . . . . . 7 0 ∈ V
22 opeq2 4376 . . . . . . . . 9 (𝑠 = 0 → ⟨𝑓, 𝑠⟩ = ⟨𝑓, 0 ⟩)
2322eqeq2d 2631 . . . . . . . 8 (𝑠 = 0 → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨𝑓, 0 ⟩))
2423anbi2d 739 . . . . . . 7 (𝑠 = 0 → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 𝑠⟩) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑌 = ⟨𝑓, 0 ⟩)))
2521, 24ceqsexv 3231 . . . . . 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 680 . . . . . 6 (((𝑓𝑇𝑌 = ⟨𝑓, 0 ⟩) ∧ (𝑅𝑓) 𝑋) ↔ (𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)))
28 an32 838 . . . . . 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 1847 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })) ↔ ∃𝑓(𝑓𝑇 ∧ (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋))))
32 elxp 5096 . . 3 (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) ↔ ∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑠 ∈ { 0 })))
33 df-rex 2913 . . 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 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  Vcvv 3189  {csn 4153  cop 4159   class class class wbr 4618  cmpt 4678   I cid 4989   × cxp 5077  cres 5081  cfv 5852  Basecbs 15788  lecple 15876  LHypclh 34777  LTrncltrn 34894  trLctrl 34952  DIsoAcdia 35824  DIsoBcdib 35934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-disoa 35825  df-dib 35935
This theorem is referenced by:  cdlemn11pre  36006  dihord2pre  36021
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