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Theorem dibelval1st 40654
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 2728 . . . . 5 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
5 eqid 2728 . . . . 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 40649 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((𝐽𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
98eleq2d 2815 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ (𝐼𝑋) ↔ 𝑌 ∈ ((𝐽𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})))
109biimp3a 1465 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → 𝑌 ∈ ((𝐽𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))
11 xp1st 8031 . 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 394  w3a 1084   = wceq 1533  wcel 2098  {csn 4632   class class class wbr 5152  cmpt 5235   I cid 5579   × cxp 5680  cres 5684  cfv 6553  1st c1st 7997  Basecbs 17187  lecple 17247  LHypclh 39489  LTrncltrn 39606  DIsoAcdia 40533  DIsoBcdib 40643
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-1st 7999  df-disoa 40534  df-dib 40644
This theorem is referenced by:  dibelval1st1  40655  dibelval1st2N  40656  diblss  40675
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