Theorem dicelval3 42503

Description: Member of the partial isomorphism C. (Contributed by NM, 26-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)

Assertion

Ref	Expression
dicelval3	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ ∃𝑠 ∈ 𝐸 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))

Distinct variable groups:   𝑔,𝑠,𝐾   𝑇,𝑔   𝑔,𝑊,𝑠   𝐸,𝑠   𝑄,𝑔,𝑠   𝑌,𝑠

Allowed substitution hints:   𝐴(𝑔, 𝑠)   𝑃(𝑔, 𝑠)   𝑇(𝑠)   𝐸(𝑔)   𝐺(𝑔, 𝑠)   𝐻(𝑔, 𝑠)   𝐼(𝑔, 𝑠)   ≤ (𝑔, 𝑠)   𝑉(𝑔, 𝑠)   𝑌(𝑔)

Proof of Theorem dicelval3

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dicval.l	. . . 4 ⊢ ≤ = (le‘𝐾)
2		dicval.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3		dicval.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4		dicval.p	. . . 4 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
5		dicval.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6		dicval.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
7		dicval.i	. . . 4 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
8		dicval2.g	. . . 4 ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
9	1, 2, 3, 4, 5, 6, 7, 8	dicval2 42502	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)})
10	9	eleq2d 2850	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}))
11		excom 2202	. . . 4 ⊢ (∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
12		an12 658	. . . . . . 7 ⊢ ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
13	12	exbii 1881	. . . . . 6 ⊢ (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
14		fvex 6929	. . . . . . 7 ⊢ (𝑠‘𝐺) ∈ V
15		opeq1 4845	. . . . . . . . 9 ⊢ (𝑓 = (𝑠‘𝐺) → ⟨𝑓, 𝑠⟩ = ⟨(𝑠‘𝐺), 𝑠⟩)
16	15	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑓 = (𝑠‘𝐺) → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
17	16	anbi1d 643	. . . . . . 7 ⊢ (𝑓 = (𝑠‘𝐺) → ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸) ↔ (𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
18	14, 17	ceqsexv 3502	. . . . . 6 ⊢ (∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)) ↔ (𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸))
19		ancom 466	. . . . . 6 ⊢ ((𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
20	13, 18, 19	3bitri 300	. . . . 5 ⊢ (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
21	20	exbii 1881	. . . 4 ⊢ (∃𝑠∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
22	11, 21	bitri 278	. . 3 ⊢ (∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
23		elopab 5508	. . 3 ⊢ (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
24		df-rex 3091	. . 3 ⊢ (∃𝑠 ∈ 𝐸 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
25	22, 23, 24	3bitr4i 306	. 2 ⊢ (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑠 ∈ 𝐸 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩)
26	10, 25	bitrdi 290	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ ∃𝑠 ∈ 𝐸 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090  ⟨cop 4840   class class class wbr 5108  {copab 5173  ‘cfv 6566  ℩crio 7407  lecple 17479  occoc 17480  Atomscatm 40586  LHypclh 41307  LTrncltrn 41424  TEndoctendo 42075  DIsoCcdic 42495

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-dic 42496

This theorem is used by:  cdlemn11pre  42533  dihord2pre  42548