Theorem dibopelval2 39159

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 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
dibval2.j	⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
dibval2.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)

Assertion

Ref	Expression
dibopelval2	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 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

Step	Hyp	Ref	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 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
6		dibval2.j	. . . 4 ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
7		dibval2.i	. . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	dibval2 39158	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 }))
9	8	eleq2d 2824	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 })))
10		opelxp 5625	. . 3 ⊢ (⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 ∈ { 0 }))
11	4	fvexi 6788	. . . . . . 7 ⊢ 𝑇 ∈ V
12	11	mptex 7099	. . . . . 6 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
13	5, 12	eqeltri 2835	. . . . 5 ⊢ 0 ∈ V
14	13	elsn2 4600	. . . 4 ⊢ (𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
15	14	anbi2i 623	. . 3 ⊢ ((𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 ∈ { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 ))
16	10, 15	bitri 274	. 2 ⊢ (⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 ))
17	9, 16	bitrdi 287	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 )))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561  ⟨cop 4567   class class class wbr 5074   ↦ cmpt 5157   I cid 5488   × cxp 5587   ↾ cres 5591  ‘cfv 6433  Basecbs 16912  lecple 16969  LHypclh 37998  LTrncltrn 38115  DIsoAcdia 39042  DIsoBcdib 39152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-disoa 39043  df-dib 39153

This theorem is referenced by:  dibopelval3  39162  dibglbN  39180  diblsmopel  39185  dib2dim  39257  dih2dimbALTN  39259  dihord6apre  39270