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Theorem diaelval 40408
Description: Member of the partial isomorphism A for a lattice 𝐾. (Contributed by NM, 3-Dec-2013.)
Hypotheses
Ref Expression
diaval.b 𝐡 = (Baseβ€˜πΎ)
diaval.l ≀ = (leβ€˜πΎ)
diaval.h 𝐻 = (LHypβ€˜πΎ)
diaval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
diaval.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
diaval.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diaelval (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (πΌβ€˜π‘‹) ↔ (𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋)))

Proof of Theorem diaelval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 diaval.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 diaval.l . . . 4 ≀ = (leβ€˜πΎ)
3 diaval.h . . . 4 𝐻 = (LHypβ€˜πΎ)
4 diaval.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 diaval.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
6 diaval.i . . . 4 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
71, 2, 3, 4, 5, 6diaval 40407 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
87eleq2d 2811 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (πΌβ€˜π‘‹) ↔ 𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋}))
9 fveq2 6882 . . . 4 (𝑓 = 𝐹 β†’ (π‘…β€˜π‘“) = (π‘…β€˜πΉ))
109breq1d 5149 . . 3 (𝑓 = 𝐹 β†’ ((π‘…β€˜π‘“) ≀ 𝑋 ↔ (π‘…β€˜πΉ) ≀ 𝑋))
1110elrab 3676 . 2 (𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋} ↔ (𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋))
128, 11bitrdi 287 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (πΌβ€˜π‘‹) ↔ (𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3424   class class class wbr 5139  β€˜cfv 6534  Basecbs 17149  lecple 17209  LHypclh 39359  LTrncltrn 39476  trLctrl 39533  DIsoAcdia 40403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-disoa 40404
This theorem is referenced by:  dian0  40414  diatrl  40419  dialss  40421  diaglbN  40430  dibelval3  40522  dibopelval3  40523  diblss  40545
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