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Theorem dibelval1st1 40547
Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.)
Hypotheses
Ref Expression
dibelval1st1.b 𝐡 = (Baseβ€˜πΎ)
dibelval1st1.l ≀ = (leβ€˜πΎ)
dibelval1st1.h 𝐻 = (LHypβ€˜πΎ)
dibelval1st1.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dibelval1st1.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dibelval1st1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘‹)) β†’ (1st β€˜π‘Œ) ∈ 𝑇)

Proof of Theorem dibelval1st1
StepHypRef Expression
1 dibelval1st1.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 dibelval1st1.l . . 3 ≀ = (leβ€˜πΎ)
3 dibelval1st1.h . . 3 𝐻 = (LHypβ€˜πΎ)
4 eqid 2727 . . 3 ((DIsoAβ€˜πΎ)β€˜π‘Š) = ((DIsoAβ€˜πΎ)β€˜π‘Š)
5 dibelval1st1.i . . 3 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5dibelval1st 40546 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘‹)) β†’ (1st β€˜π‘Œ) ∈ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘‹))
7 dibelval1st1.t . . 3 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
81, 2, 3, 7, 4diael 40440 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ (1st β€˜π‘Œ) ∈ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜π‘‹)) β†’ (1st β€˜π‘Œ) ∈ 𝑇)
96, 8syld3an3 1407 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘‹)) β†’ (1st β€˜π‘Œ) ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5142  β€˜cfv 6542  1st c1st 7983  Basecbs 17165  lecple 17225  LHypclh 39381  LTrncltrn 39498  DIsoAcdia 40425  DIsoBcdib 40535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1st 7985  df-disoa 40426  df-dib 40536
This theorem is referenced by:  diblss  40567
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