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Theorem dibelval1st 38165
Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.)
Hypotheses
Ref Expression
dibelval1.b 𝐵 = (Base‘𝐾)
dibelval1.l = (le‘𝐾)
dibelval1.h 𝐻 = (LHyp‘𝐾)
dibelval1.j 𝐽 = ((DIsoA‘𝐾)‘𝑊)
dibelval1.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibelval1st (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (1st𝑌) ∈ (𝐽𝑋))

Proof of Theorem dibelval1st
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 dibelval1.b . . . . 5 𝐵 = (Base‘𝐾)
2 dibelval1.l . . . . 5 = (le‘𝐾)
3 dibelval1.h . . . . 5 𝐻 = (LHyp‘𝐾)
4 eqid 2818 . . . . 5 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
5 eqid 2818 . . . . 5 (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
6 dibelval1.j . . . . 5 𝐽 = ((DIsoA‘𝐾)‘𝑊)
7 dibelval1.i . . . . 5 𝐼 = ((DIsoB‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dibval2 38160 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((𝐽𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
98eleq2d 2895 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ (𝐼𝑋) ↔ 𝑌 ∈ ((𝐽𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})))
109biimp3a 1460 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → 𝑌 ∈ ((𝐽𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
11 xp1st 7710 . 2 (𝑌 ∈ ((𝐽𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) → (1st𝑌) ∈ (𝐽𝑋))
1210, 11syl 17 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (1st𝑌) ∈ (𝐽𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  {csn 4557   class class class wbr 5057  cmpt 5137   I cid 5452   × cxp 5546  cres 5550  cfv 6348  1st c1st 7676  Basecbs 16471  lecple 16560  LHypclh 37000  LTrncltrn 37117  DIsoAcdia 38044  DIsoBcdib 38154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-1st 7678  df-disoa 38045  df-dib 38155
This theorem is referenced by:  dibelval1st1  38166  dibelval1st2N  38167  diblss  38186
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