Theorem dicopelval2 38321

Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 20-Feb-2014.)

Hypotheses

Ref	Expression
dicval.l	⊢ ≤ = (le‘𝐾)
dicval.a	⊢ 𝐴 = (Atoms‘𝐾)
dicval.h	⊢ 𝐻 = (LHyp‘𝐾)
dicval.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
dicval.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dicval.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dicval.i	⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
dicval2.g	⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
dicelval2.f	⊢ 𝐹 ∈ V
dicelval2.s	⊢ 𝑆 ∈ V

Assertion

Ref	Expression
dicopelval2	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑄) ↔ (𝐹 = (𝑆‘𝐺) ∧ 𝑆 ∈ 𝐸)))

Distinct variable groups:   𝑔,𝐾   𝑇,𝑔   𝑔,𝑊   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

Step	Hyp	Ref	Expression
1		dicval.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		dicval.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		dicval.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		dicval.p	. . 3 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
5		dicval.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6		dicval.e	. . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
7		dicval.i	. . 3 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
8		dicelval2.f	. . 3 ⊢ 𝐹 ∈ V
9		dicelval2.s	. . 3 ⊢ 𝑆 ∈ V
10	1, 2, 3, 4, 5, 6, 7, 8, 9	dicopelval 38317	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑄) ↔ (𝐹 = (𝑆‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑆 ∈ 𝐸)))
11		dicval2.g	. . . . 5 ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
12	11	fveq2i 6676	. . . 4 ⊢ (𝑆‘𝐺) = (𝑆‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))
13	12	eqeq2i 2837	. . 3 ⊢ (𝐹 = (𝑆‘𝐺) ↔ 𝐹 = (𝑆‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
14	13	anbi1i 625	. 2 ⊢ ((𝐹 = (𝑆‘𝐺) ∧ 𝑆 ∈ 𝐸) ↔ (𝐹 = (𝑆‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑆 ∈ 𝐸))
15	10, 14	syl6bbr 291	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑄) ↔ (𝐹 = (𝑆‘𝐺) ∧ 𝑆 ∈ 𝐸)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  Vcvv 3497  ⟨cop 4576   class class class wbr 5069  ‘cfv 6358  ℩crio 7116  lecple 16575  occoc 16576  Atomscatm 36403  LHypclh 37124  LTrncltrn 37241  TEndoctendo 37892  DIsoCcdic 38312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-dic 38313

This theorem is referenced by:  diclspsn  38334  cdlemn11a  38347  dihopelvalcqat  38386  dihopelvalcpre  38388  dihord6apre  38396