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Theorem diaelval 39546
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 39545 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
87eleq2d 2820 . 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 6846 . . . 4 (𝑓 = 𝐹 β†’ (π‘…β€˜π‘“) = (π‘…β€˜πΉ))
109breq1d 5119 . . 3 (𝑓 = 𝐹 β†’ ((π‘…β€˜π‘“) ≀ 𝑋 ↔ (π‘…β€˜πΉ) ≀ 𝑋))
1110elrab 3649 . 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 397   = wceq 1542   ∈ wcel 2107  {crab 3406   class class class wbr 5109  β€˜cfv 6500  Basecbs 17091  lecple 17148  LHypclh 38497  LTrncltrn 38614  trLctrl 38671  DIsoAcdia 39541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-disoa 39542
This theorem is referenced by:  dian0  39552  diatrl  39557  dialss  39559  diaglbN  39568  dibelval3  39660  dibopelval3  39661  diblss  39683
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