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Theorem dibopelvalN 38351
Description: Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibval.b 𝐵 = (Base‘𝐾)
dibval.h 𝐻 = (LHyp‘𝐾)
dibval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibval.o 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
dibval.j 𝐽 = ((DIsoA‘𝐾)‘𝑊)
dibval.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibopelvalN (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 )))
Distinct variable groups:   𝑓,𝐾   𝑓,𝑊   𝑇,𝑓
Allowed substitution hints:   𝐵(𝑓)   𝑆(𝑓)   𝐹(𝑓)   𝐻(𝑓)   𝐼(𝑓)   𝐽(𝑓)   𝑉(𝑓)   𝑋(𝑓)   0 (𝑓)

Proof of Theorem dibopelvalN
StepHypRef Expression
1 dibval.b . . . 4 𝐵 = (Base‘𝐾)
2 dibval.h . . . 4 𝐻 = (LHyp‘𝐾)
3 dibval.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 dibval.o . . . 4 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
5 dibval.j . . . 4 𝐽 = ((DIsoA‘𝐾)‘𝑊)
6 dibval.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
71, 2, 3, 4, 5, 6dibval 38350 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
87eleq2d 2901 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 })))
9 opelxp 5579 . . 3 (⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 ∈ { 0 }))
103fvexi 6673 . . . . . . 7 𝑇 ∈ V
1110mptex 6975 . . . . . 6 (𝑓𝑇 ↦ ( I ↾ 𝐵)) ∈ V
124, 11eqeltri 2912 . . . . 5 0 ∈ V
1312elsn2 4589 . . . 4 (𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
1413anbi2i 625 . . 3 ((𝐹 ∈ (𝐽𝑋) ∧ 𝑆 ∈ { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 ))
159, 14bitri 278 . 2 (⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 ))
168, 15syl6bb 290 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  {csn 4550  cop 4556  cmpt 5133   I cid 5447   × cxp 5541  dom cdm 5543  cres 5545  cfv 6344  Basecbs 16481  LHypclh 37192  LTrncltrn 37309  DIsoAcdia 38236  DIsoBcdib 38346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-dib 38347
This theorem is referenced by: (None)
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