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Theorem dibopelval2 38460
 Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.)
Hypotheses
Ref Expression
dibval2.b 𝐵 = (Base‘𝐾)
dibval2.l = (le‘𝐾)
dibval2.h 𝐻 = (LHyp‘𝐾)
dibval2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibval2.o 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
dibval2.j 𝐽 = ((DIsoA‘𝐾)‘𝑊)
dibval2.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibopelval2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨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)   (𝑓)   𝑉(𝑓)   𝑋(𝑓)   0 (𝑓)

Proof of Theorem dibopelval2
StepHypRef Expression
1 dibval2.b . . . 4 𝐵 = (Base‘𝐾)
2 dibval2.l . . . 4 = (le‘𝐾)
3 dibval2.h . . . 4 𝐻 = (LHyp‘𝐾)
4 dibval2.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
5 dibval2.o . . . 4 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
6 dibval2.j . . . 4 𝐽 = ((DIsoA‘𝐾)‘𝑊)
7 dibval2.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dibval2 38459 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
98eleq2d 2875 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 })))
10 opelxp 5556 . . 3 (⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 ∈ { 0 }))
114fvexi 6660 . . . . . . 7 𝑇 ∈ V
1211mptex 6964 . . . . . 6 (𝑓𝑇 ↦ ( I ↾ 𝐵)) ∈ V
135, 12eqeltri 2886 . . . . 5 0 ∈ V
1413elsn2 4564 . . . 4 (𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
1514anbi2i 625 . . 3 ((𝐹 ∈ (𝐽𝑋) ∧ 𝑆 ∈ { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 ))
1610, 15bitri 278 . 2 (⟨𝐹, 𝑆⟩ ∈ ((𝐽𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 ))
179, 16syl6bb 290 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {csn 4525  ⟨cop 4531   class class class wbr 5031   ↦ cmpt 5111   I cid 5425   × cxp 5518   ↾ cres 5522  ‘cfv 6325  Basecbs 16478  lecple 16567  LHypclh 37299  LTrncltrn 37416  DIsoAcdia 38343  DIsoBcdib 38453 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-disoa 38344  df-dib 38454 This theorem is referenced by:  dibopelval3  38463  dibglbN  38481  diblsmopel  38486  dib2dim  38558  dih2dimbALTN  38560  dihord6apre  38571
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