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Theorem dibopelvalN 35450
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 35449 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
87eleq2d 2673 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 })))
9 opelxp 5070 . . 3 (⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 ∈ { 0 }))
10 fvex 6113 . . . . . . . 8 ((LTrn‘𝐾)‘𝑊) ∈ V
113, 10eqeltri 2684 . . . . . . 7 𝑇 ∈ V
1211mptex 6390 . . . . . 6 (𝑓𝑇 ↦ ( I ↾ 𝐵)) ∈ V
134, 12eqeltri 2684 . . . . 5 0 ∈ V
1413elsn2 4158 . . . 4 (𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
1514anbi2i 726 . . 3 ((𝐹 ∈ (𝐽𝑋) ∧ 𝑆 ∈ { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 ))
169, 15bitri 263 . 2 (⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 ))
178, 16syl6bb 275 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  {csn 4125  cop 4131  cmpt 4643   I cid 4948   × cxp 5036  dom cdm 5038  cres 5040  cfv 5804  Basecbs 15695  LHypclh 34288  LTrncltrn 34405  DIsoAcdia 35335  DIsoBcdib 35445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-dib 35446
This theorem is referenced by: (None)
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