Theorem dicelval3 40515

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 40514	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)})
10	9	eleq2d 2818	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}))
11		excom 2161	. . . 4 ⊢ (∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
12		an12 642	. . . . . . 7 ⊢ ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
13	12	exbii 1849	. . . . . 6 ⊢ (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
14		fvex 6904	. . . . . . 7 ⊢ (𝑠‘𝐺) ∈ V
15		opeq1 4873	. . . . . . . . 9 ⊢ (𝑓 = (𝑠‘𝐺) → ⟨𝑓, 𝑠⟩ = ⟨(𝑠‘𝐺), 𝑠⟩)
16	15	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑓 = (𝑠‘𝐺) → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
17	16	anbi1d 629	. . . . . . 7 ⊢ (𝑓 = (𝑠‘𝐺) → ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸) ↔ (𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
18	14, 17	ceqsexv 3525	. . . . . 6 ⊢ (∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)) ↔ (𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸))
19		ancom 460	. . . . . 6 ⊢ ((𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
20	13, 18, 19	3bitri 297	. . . . 5 ⊢ (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
21	20	exbii 1849	. . . 4 ⊢ (∃𝑠∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
22	11, 21	bitri 275	. . 3 ⊢ (∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
23		elopab 5527	. . 3 ⊢ (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
24		df-rex 3070	. . 3 ⊢ (∃𝑠 ∈ 𝐸 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
25	22, 23, 24	3bitr4i 303	. 2 ⊢ (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑠 ∈ 𝐸 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩)
26	10, 25	bitrdi 287	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 205   ∧ wa 395   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ∃wrex 3069  ⟨cop 4634   class class class wbr 5148  {copab 5210  ‘cfv 6543  ℩crio 7367  lecple 17211  occoc 17212  Atomscatm 38597  LHypclh 39319  LTrncltrn 39436  TEndoctendo 40087  DIsoCcdic 40507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-dic 40508

This theorem is referenced by:  cdlemn11pre  40545  dihord2pre  40560