Theorem diaelval 41496

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 41495	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})
8	7	eleq2d 2823	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ 𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}))
9		fveq2 6835	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑅‘𝑓) = (𝑅‘𝐹))
10	9	breq1d 5096	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑅‘𝑓) ≤ 𝑋 ↔ (𝑅‘𝐹) ≤ 𝑋))
11	10	elrab 3635	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋))
12	8, 11	bitrdi 287	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3390   class class class wbr 5086  ‘cfv 6493  Basecbs 17173  lecple 17221  LHypclh 40447  LTrncltrn 40564  trLctrl 40621  DIsoAcdia 41491

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-disoa 41492

This theorem is referenced by:  dian0  41502  diatrl  41507  dialss  41509  diaglbN  41518  dibelval3  41610  dibopelval3  41611  diblss  41633