Theorem diaval 41051

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 41050	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))
8	7	adantr 480	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))
9	8	fveq1d 6878	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})‘𝑋))
10		simpr 484	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
11		breq1 5122	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
12	11	elrab 3671	. . . 4 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
13	10, 12	sylibr 234	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊})
14		breq2 5123	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑅‘𝑓) ≤ 𝑥 ↔ (𝑅‘𝑓) ≤ 𝑋))
15	14	rabbidv 3423	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})
16		eqid 2735	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}) = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})
17	4	fvexi 6890	. . . . 5 ⊢ 𝑇 ∈ V
18	17	rabex 5309	. . . 4 ⊢ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} ∈ V
19	15, 16, 18	fvmpt 6986	. . 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 2770	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {crab 3415   class class class wbr 5119   ↦ cmpt 5201  ‘cfv 6531  Basecbs 17228  lecple 17278  LHypclh 40003  LTrncltrn 40120  trLctrl 40177  DIsoAcdia 41047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-disoa 41048

This theorem is referenced by:  diaelval  41052  diass  41061  diaord  41066  dia0  41071  dia1N  41072  diassdvaN  41079  dia1dim  41080  cdlemm10N  41137  dibval3N  41165  dihwN  41308