Theorem diaval 41905

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 41904	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))
8	7	adantr 486	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))
9	8	fveq1d 6880	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})‘𝑋))
10		breq1 5106	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
11	10	elrab 3645	. . . 4 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
12	11	bilanri 512	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊})
13		breq2 5107	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑅‘𝑓) ≤ 𝑥 ↔ (𝑅‘𝑓) ≤ 𝑋))
14	13	rabbidv 3419	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})
15		eqid 2760	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}) = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})
16	4	fvexi 6892	. . . . 5 ⊢ 𝑇 ∈ V
17	16	rabex 5303	. . . 4 ⊢ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} ∈ V
18	14, 15, 17	fvmpt 6986	. . 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 2795	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6533  Basecbs 17301  lecple 17349  LHypclh 40857  LTrncltrn 40974  trLctrl 41031  DIsoAcdia 41901

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-disoa 41902

This theorem is used by:  diaelval  41906  diass  41915  diaord  41920  dia0  41925  dia1N  41926  diassdvaN  41933  dia1dim  41934  cdlemm10N  41991  dibval3N  42019  dihwN  42162