Theorem dicelval3 37201

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 37200	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)})
10	9	eleq2d 2864	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}))
11		excom 2205	. . . 4 ⊢ (∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
12		an12 636	. . . . . . 7 ⊢ ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
13	12	exbii 1944	. . . . . 6 ⊢ (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
14		fvex 6424	. . . . . . 7 ⊢ (𝑠‘𝐺) ∈ V
15		opeq1 4593	. . . . . . . . 9 ⊢ (𝑓 = (𝑠‘𝐺) → ⟨𝑓, 𝑠⟩ = ⟨(𝑠‘𝐺), 𝑠⟩)
16	15	eqeq2d 2809	. . . . . . . 8 ⊢ (𝑓 = (𝑠‘𝐺) → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
17	16	anbi1d 624	. . . . . . 7 ⊢ (𝑓 = (𝑠‘𝐺) → ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸) ↔ (𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸)))
18	14, 17	ceqsexv 3430	. . . . . 6 ⊢ (∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠 ∈ 𝐸)) ↔ (𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸))
19		ancom 453	. . . . . 6 ⊢ ((𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑠 ∈ 𝐸) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
20	13, 18, 19	3bitri 289	. . . . 5 ⊢ (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
21	20	exbii 1944	. . . 4 ⊢ (∃𝑠∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
22	11, 21	bitri 267	. . 3 ⊢ (∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
23		elopab 5179	. . 3 ⊢ (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑓∃𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)))
24		df-rex 3095	. . 3 ⊢ (∃𝑠 ∈ 𝐸 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩ ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩))
25	22, 23, 24	3bitr4i 295	. 2 ⊢ (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑠 ∈ 𝐸 𝑌 = ⟨(𝑠‘𝐺), 𝑠⟩)
26	10, 25	syl6bb 279	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 198   ∧ wa 385   = wceq 1653  ∃wex 1875   ∈ wcel 2157  ∃wrex 3090  ⟨cop 4374   class class class wbr 4843  {copab 4905  ‘cfv 6101  ℩crio 6838  lecple 16274  occoc 16275  Atomscatm 35284  LHypclh 36005  LTrncltrn 36122  TEndoctendo 36773  DIsoCcdic 37193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-dic 37194

This theorem is referenced by:  cdlemn11pre  37231  dihord2pre  37246