Theorem dibopelval2 41652

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 41651	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 }))
9	8	eleq2d 2827	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 })))
10		opelxp 5657	. . 3 ⊢ (⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 ∈ { 0 }))
11	4	fvexi 6845	. . . . . . 7 ⊢ 𝑇 ∈ V
12	11	mptex 7171	. . . . . 6 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
13	5, 12	eqeltri 2837	. . . . 5 ⊢ 0 ∈ V
14	13	elsn2 4600	. . . 4 ⊢ (𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
15	14	anbi2i 630	. . 3 ⊢ ((𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 ∈ { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 ))
16	10, 15	bitri 277	. 2 ⊢ (⟨𝐹, 𝑆⟩ ∈ ((𝐽‘𝑋) × { 0 }) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 ))
17	9, 16	bitrdi 289	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 )))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Vcvv 3433  {csn 4558  ⟨cop 4564   class class class wbr 5075   ↦ cmpt 5156   I cid 5515   × cxp 5619   ↾ cres 5623  ‘cfv 6489  Basecbs 17174  lecple 17222  LHypclh 40491  LTrncltrn 40608  DIsoAcdia 41535  DIsoBcdib 41645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-disoa 41536  df-dib 41646

This theorem is referenced by:  dibopelval3  41655  dibglbN  41673  diblsmopel  41678  dib2dim  41750  dih2dimbALTN  41752  dihord6apre  41763