Theorem diaval 41691

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 41690	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))
8	7	adantr 485	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))
9	8	fveq1d 6881	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})‘𝑋))
10		breq1 5113	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
11	10	elrab 3659	. . . 4 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
12	11	bilanri 511	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊})
13		breq2 5114	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑅‘𝑓) ≤ 𝑥 ↔ (𝑅‘𝑓) ≤ 𝑋))
14	13	rabbidv 3430	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})
15		eqid 2769	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}) = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})
16	4	fvexi 6893	. . . . 5 ⊢ 𝑇 ∈ V
17	16	rabex 5307	. . . 4 ⊢ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} ∈ V
18	14, 15, 17	fvmpt 6987	. . 3 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} → ((𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})
19	12, 18	syl 18	. 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})
20	9, 19	eqtrd 2804	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {crab 3423   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6534  Basecbs 17265  lecple 17313  LHypclh 40643  LTrncltrn 40760  trLctrl 40817  DIsoAcdia 41687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-disoa 41688

This theorem is referenced by:  diaelval  41692  diass  41701  diaord  41706  dia0  41711  dia1N  41712  diassdvaN  41719  dia1dim  41720  cdlemm10N  41777  dibval3N  41805  dihwN  41948