Theorem dicelval3 35487

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 35486	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)})
10	9	eleq2d 2673	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}))
11		excom 2029	. . . 4 ⊢ (∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
12		an12 834	. . . . . . 7 ⊢ ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
13	12	exbii 1764	. . . . . 6 ⊢ (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
14		fvex 6113	. . . . . . 7 ⊢ (𝑠‘𝐺) ∈ V
15		opeq1 4340	. . . . . . . . 9 ⊢ (𝑓 = (𝑠‘𝐺) → ⟨𝑓, 𝑠⟩ = ⟨(𝑠‘𝐺), 𝑠⟩)
16	15	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑓 = (𝑠‘𝐺) → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
17	16	anbi1d 737	. . . . . . 7 ⊢ (𝑓 = (𝑠‘𝐺) → ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸) ↔ (𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
18	14, 17	ceqsexv 3215	. . . . . 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 285	. . . . 5 ⊢ (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
21	20	exbii 1764	. . . 4 ⊢ (∃𝑠∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
22	11, 21	bitri 263	. . 3 ⊢ (∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
23		elopab 4908	. . 3 ⊢ (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
24		df-rex 2902	. . 3 ⊢ (∃𝑠 ∈ 𝐸 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
25	22, 23, 24	3bitr4i 291	. 2 ⊢ (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑠 ∈ 𝐸 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩)
26	10, 25	syl6bb 275	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 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  ⟨cop 4131   class class class wbr 4583  {copab 4642  ‘cfv 5804  ℩crio 6510  lecple 15775  occoc 15776  Atomscatm 33568  LHypclh 34288  LTrncltrn 34405  TEndoctendo 35058  DIsoCcdic 35479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-dic 35480

This theorem is referenced by:  cdlemn11pre  35517  dihord2pre  35532