Theorem dibopelval2 41139

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 41138	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 }))
9	8	eleq2d 2814	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 })))
10		opelxp 5674	. . 3 ⊢ (⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 ∈ { 0 }))
11	4	fvexi 6872	. . . . . . 7 ⊢ 𝑇 ∈ V
12	11	mptex 7197	. . . . . 6 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
13	5, 12	eqeltri 2824	. . . . 5 ⊢ 0 ∈ V
14	13	elsn2 4629	. . . 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 1540   ∈ wcel 2109  Vcvv 3447  {csn 4589  ⟨cop 4595   class class class wbr 5107   ↦ cmpt 5188   I cid 5532   × cxp 5636   ↾ cres 5640  ‘cfv 6511  Basecbs 17179  lecple 17227  LHypclh 39978  LTrncltrn 40095  DIsoAcdia 41022  DIsoBcdib 41132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-disoa 41023  df-dib 41133

This theorem is referenced by:  dibopelval3  41142  dibglbN  41160  diblsmopel  41165  dib2dim  41237  dih2dimbALTN  41239  dihord6apre  41250