Theorem dicvalrelN 42243

Description: The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dicvalrel.h	⊢ 𝐻 = (LHyp‘𝐾)
dicvalrel.i	⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)

Assertion

Ref	Expression
dicvalrelN	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑋))

Proof of Theorem dicvalrelN

Dummy variables 𝑓 𝑔 𝑝 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopabv 5798	. . . 4 ⊢ Rel {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}
2		eqid 2760	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2760	. . . . . . . . . 10 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		dicvalrel.h	. . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾)
5		dicvalrel.i	. . . . . . . . . 10 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
6	2, 3, 4, 5	dicdmN 42242	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊})
7	6	eleq2d 2846	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊}))
8		breq1 5105	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → (𝑝(le‘𝐾)𝑊 ↔ 𝑋(le‘𝐾)𝑊))
9	8	notbid 321	. . . . . . . . 9 ⊢ (𝑝 = 𝑋 → (¬ 𝑝(le‘𝐾)𝑊 ↔ ¬ 𝑋(le‘𝐾)𝑊))
10	9	elrab 3644	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊} ↔ (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊))
11	7, 10	bitrdi 290	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊)))
12	11	biimpa 482	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊))
13		eqid 2760	. . . . . . 7 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
14		eqid 2760	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
15		eqid 2760	. . . . . . 7 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
16	2, 3, 4, 13, 14, 15, 5	dicval 42234	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊)) → (𝐼‘𝑋) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
17	12, 16	syldan 603	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
18	17	releqd 5754	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (Rel (𝐼‘𝑋) ↔ Rel {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}))
19	1, 18	mpbiri 261	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → Rel (𝐼‘𝑋))
20	19	ex 418	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 → Rel (𝐼‘𝑋)))
21		rel0 5775	. . 3 ⊢ Rel ∅
22		ndmfv 6917	. . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (𝐼‘𝑋) = ∅)
23	22	releqd 5754	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (Rel (𝐼‘𝑋) ↔ Rel ∅))
24	21, 23	mpbiri 261	. 2 ⊢ (¬ 𝑋 ∈ dom 𝐼 → Rel (𝐼‘𝑋))
25	20, 24	pm2.61d1 182	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑋))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {crab 3412  ∅c0 4278   class class class wbr 5102  {copab 5166  dom cdm 5650  Rel wrel 5655  ‘cfv 6538  ℩crio 7376  lecple 17435  occoc 17436  Atomscatm 40321  LHypclh 41042  LTrncltrn 41159  TEndoctendo 41810  DIsoCcdic 42230

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-dic 42231

This theorem is used by: (None)