Theorem dibopelval2 41128

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 41127	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 }))
9	8	eleq2d 2825	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 })))
10		opelxp 5725	. . 3 ⊢ (⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 ∈ { 0 }))
11	4	fvexi 6921	. . . . . . 7 ⊢ 𝑇 ∈ V
12	11	mptex 7243	. . . . . 6 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
13	5, 12	eqeltri 2835	. . . . 5 ⊢ 0 ∈ V
14	13	elsn2 4670	. . . 4 ⊢ (𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
15	14	anbi2i 623	. . 3 ⊢ ((𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 ∈ { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 ))
16	10, 15	bitri 275	. 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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  {csn 4631  ⟨cop 4637   class class class wbr 5148   ↦ cmpt 5231   I cid 5582   × cxp 5687   ↾ cres 5691  ‘cfv 6563  Basecbs 17245  lecple 17305  LHypclh 39967  LTrncltrn 40084  DIsoAcdia 41011  DIsoBcdib 41121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-disoa 41012  df-dib 41122

This theorem is referenced by:  dibopelval3  41131  dibglbN  41149  diblsmopel  41154  dib2dim  41226  dih2dimbALTN  41228  dihord6apre  41239