Theorem diaelval 41525

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 41524	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})
8	7	eleq2d 2825	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ 𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}))
9		fveq2 6827	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑅‘𝑓) = (𝑅‘𝐹))
10	9	breq1d 5082	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑅‘𝑓) ≤ 𝑋 ↔ (𝑅‘𝐹) ≤ 𝑋))
11	10	elrab 3629	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋))
12	8, 11	bitrdi 288	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3391   class class class wbr 5072  ‘cfv 6485  Basecbs 17170  lecple 17218  LHypclh 40476  LTrncltrn 40593  trLctrl 40650  DIsoAcdia 41520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-disoa 41521

This theorem is referenced by:  dian0  41531  diatrl  41536  dialss  41538  diaglbN  41547  dibelval3  41639  dibopelval3  41640  diblss  41662