Theorem dicelval3 41839

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 41838	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)})
10	9	eleq2d 2855	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}))
11		excom 2203	. . . 4 ⊢ (∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
12		an12 657	. . . . . . 7 ⊢ ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
13	12	exbii 1875	. . . . . 6 ⊢ (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
14		fvex 6892	. . . . . . 7 ⊢ (𝑠‘𝐺) ∈ V
15		opeq1 4839	. . . . . . . . 9 ⊢ (𝑓 = (𝑠‘𝐺) → ⟨𝑓, 𝑠⟩ = ⟨(𝑠‘𝐺), 𝑠⟩)
16	15	eqeq2d 2780	. . . . . . . 8 ⊢ (𝑓 = (𝑠‘𝐺) → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
17	16	anbi1d 642	. . . . . . 7 ⊢ (𝑓 = (𝑠‘𝐺) → ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸) ↔ (𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
18	14, 17	ceqsexv 3511	. . . . . 6 ⊢ (∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)) ↔ (𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸))
19		ancom 465	. . . . . 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 1875	. . . 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 5509	. . 3 ⊢ (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
24		df-rex 3096	. . 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⟩))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃wrex 3095  ⟨cop 4597   class class class wbr 5110  {copab 5174  ‘cfv 6533  ℩crio 7364  lecple 17313  occoc 17314  Atomscatm 39922  LHypclh 40643  LTrncltrn 40760  TEndoctendo 41411  DIsoCcdic 41831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-dic 41832

This theorem is referenced by:  cdlemn11pre  41869  dihord2pre  41884