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Theorem dicopelval2 40709
Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 20-Feb-2014.)
Hypotheses
Ref Expression
dicval.l ≀ = (leβ€˜πΎ)
dicval.a 𝐴 = (Atomsβ€˜πΎ)
dicval.h 𝐻 = (LHypβ€˜πΎ)
dicval.p 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
dicval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dicval.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dicval.i 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
dicval2.g 𝐺 = (℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)
dicelval2.f 𝐹 ∈ V
dicelval2.s 𝑆 ∈ V
Assertion
Ref Expression
dicopelval2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (⟨𝐹, π‘†βŸ© ∈ (πΌβ€˜π‘„) ↔ (𝐹 = (π‘†β€˜πΊ) ∧ 𝑆 ∈ 𝐸)))
Distinct variable groups:   𝑔,𝐾   𝑇,𝑔   𝑔,π‘Š   𝑄,𝑔
Allowed substitution hints:   𝐴(𝑔)   𝑃(𝑔)   𝑆(𝑔)   𝐸(𝑔)   𝐹(𝑔)   𝐺(𝑔)   𝐻(𝑔)   𝐼(𝑔)   ≀ (𝑔)   𝑉(𝑔)

Proof of Theorem dicopelval2
StepHypRef Expression
1 dicval.l . . 3 ≀ = (leβ€˜πΎ)
2 dicval.a . . 3 𝐴 = (Atomsβ€˜πΎ)
3 dicval.h . . 3 𝐻 = (LHypβ€˜πΎ)
4 dicval.p . . 3 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
5 dicval.t . . 3 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
6 dicval.e . . 3 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
7 dicval.i . . 3 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
8 dicelval2.f . . 3 𝐹 ∈ V
9 dicelval2.s . . 3 𝑆 ∈ V
101, 2, 3, 4, 5, 6, 7, 8, 9dicopelval 40705 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (⟨𝐹, π‘†βŸ© ∈ (πΌβ€˜π‘„) ↔ (𝐹 = (π‘†β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑆 ∈ 𝐸)))
11 dicval2.g . . . . 5 𝐺 = (℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)
1211fveq2i 6894 . . . 4 (π‘†β€˜πΊ) = (π‘†β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄))
1312eqeq2i 2738 . . 3 (𝐹 = (π‘†β€˜πΊ) ↔ 𝐹 = (π‘†β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)))
1413anbi1i 622 . 2 ((𝐹 = (π‘†β€˜πΊ) ∧ 𝑆 ∈ 𝐸) ↔ (𝐹 = (π‘†β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑆 ∈ 𝐸))
1510, 14bitr4di 288 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (⟨𝐹, π‘†βŸ© ∈ (πΌβ€˜π‘„) ↔ (𝐹 = (π‘†β€˜πΊ) ∧ 𝑆 ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463  βŸ¨cop 4630   class class class wbr 5143  β€˜cfv 6542  β„©crio 7370  lecple 17237  occoc 17238  Atomscatm 38790  LHypclh 39512  LTrncltrn 39629  TEndoctendo 40280  DIsoCcdic 40700
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  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 5144  df-opab 5206  df-mpt 5227  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-riota 7371  df-dic 40701
This theorem is referenced by:  diclspsn  40722  cdlemn11a  40735  dihopelvalcqat  40774  dihopelvalcpre  40776  dihord6apre  40784
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