Theorem dicvalrelN 42242

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 5799	. . . 4 ⊢ Rel {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}
2		eqid 2761	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2761	. . . . . . . . . 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 42241	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊})
7	6	eleq2d 2847	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊}))
8		breq1 5106	. . . . . . . . . 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 3645	. . . . . . . 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 2761	. . . . . . 7 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
14		eqid 2761	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
15		eqid 2761	. . . . . . 7 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
16	2, 3, 4, 13, 14, 15, 5	dicval 42233	. . . . . 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 5755	. . . 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 5776	. . 3 ⊢ Rel ∅
22		ndmfv 6917	. . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (𝐼‘𝑋) = ∅)
23	22	releqd 5755	. . 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 1570   ∈ wcel 2145  {crab 3413  ∅c0 4279   class class class wbr 5103  {copab 5167  dom cdm 5651  Rel wrel 5656  ‘cfv 6538  ℩crio 7376  lecple 17435  occoc 17436  Atomscatm 40320  LHypclh 41041  LTrncltrn 41158  TEndoctendo 41809  DIsoCcdic 42229

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 42230

This theorem is used by: (None)