Theorem diaelval 39899

Description: Member of the partial isomorphism A for a lattice 𝐾. (Contributed by NM, 3-Dec-2013.)

Hypotheses

Ref	Expression
diaval.b	⊢ 𝐵 = (Base‘𝐾)
diaval.l	⊢ ≤ = (le‘𝐾)
diaval.h	⊢ 𝐻 = (LHyp‘𝐾)
diaval.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
diaval.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
diaval.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)

Assertion

Ref	Expression
diaelval	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋)))

Proof of Theorem diaelval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		diaval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		diaval.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		diaval.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4		diaval.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5		diaval.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
6		diaval.i	. . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
7	1, 2, 3, 4, 5, 6	diaval 39898	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})
8	7	eleq2d 2819	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ 𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}))
9		fveq2 6891	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑅‘𝑓) = (𝑅‘𝐹))
10	9	breq1d 5158	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑅‘𝑓) ≤ 𝑋 ↔ (𝑅‘𝐹) ≤ 𝑋))
11	10	elrab 3683	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋))
12	8, 11	bitrdi 286	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432   class class class wbr 5148  ‘cfv 6543  Basecbs 17143  lecple 17203  LHypclh 38850  LTrncltrn 38967  trLctrl 39024  DIsoAcdia 39894

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-disoa 39895

This theorem is referenced by:  dian0  39905  diatrl  39910  dialss  39912  diaglbN  39921  dibelval3  40013  dibopelval3  40014  diblss  40036