Theorem diaval 38783

Description: The partial isomorphism A for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-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
diaval	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})

Distinct variable groups:   𝑓,𝐾   𝑇,𝑓   𝑓,𝑊   𝑓,𝑋

Allowed substitution hints:   𝐵(𝑓)   𝑅(𝑓)   𝐻(𝑓)   𝐼(𝑓)   ≤ (𝑓)   𝑉(𝑓)

Proof of Theorem diaval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		diaval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		diaval.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		diaval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		diaval.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5		diaval.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
6		diaval.i	. . . . 5 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
7	1, 2, 3, 4, 5, 6	diafval 38782	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))
8	7	adantr 484	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))
9	8	fveq1d 6719	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})‘𝑋))
10		simpr 488	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
11		breq1 5056	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
12	11	elrab 3602	. . . 4 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
13	10, 12	sylibr 237	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊})
14		breq2 5057	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑅‘𝑓) ≤ 𝑥 ↔ (𝑅‘𝑓) ≤ 𝑋))
15	14	rabbidv 3390	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})
16		eqid 2737	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}) = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})
17	4	fvexi 6731	. . . . 5 ⊢ 𝑇 ∈ V
18	17	rabex 5225	. . . 4 ⊢ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} ∈ V
19	15, 16, 18	fvmpt 6818	. . 3 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} → ((𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})
20	13, 19	syl 17	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})
21	9, 20	eqtrd 2777	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  {crab 3065   class class class wbr 5053   ↦ cmpt 5135  ‘cfv 6380  Basecbs 16760  lecple 16809  LHypclh 37735  LTrncltrn 37852  trLctrl 37909  DIsoAcdia 38779

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 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-disoa 38780

This theorem is referenced by:  diaelval  38784  diass  38793  diaord  38798  dia0  38803  dia1N  38804  diassdvaN  38811  dia1dim  38812  cdlemm10N  38869  dibval3N  38897  dihwN  39040