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Theorem dibopelvalN 41125
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 41124 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
87eleq2d 2824 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 })))
9 opelxp 5724 . . 3 (⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 ∈ { 0 }))
103fvexi 6920 . . . . . . 7 𝑇 ∈ V
1110mptex 7242 . . . . . 6 (𝑓𝑇 ↦ ( I ↾ 𝐵)) ∈ V
124, 11eqeltri 2834 . . . . 5 0 ∈ V
1312elsn2 4669 . . . 4 (𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
1413anbi2i 623 . . 3 ((𝐹 ∈ (𝐽𝑋) ∧ 𝑆 ∈ { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 ))
159, 14bitri 275 . 2 (⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 ))
168, 15bitrdi 287 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  {csn 4630  cop 4636  cmpt 5230   I cid 5581   × cxp 5686  dom cdm 5688  cres 5690  cfv 6562  Basecbs 17244  LHypclh 39966  LTrncltrn 40083  DIsoAcdia 41010  DIsoBcdib 41120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-dib 41121
This theorem is referenced by: (None)
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