Theorem dicvalrelN 41467

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 5770	. . . 4 ⊢ Rel {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑋)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}
2		eqid 2736	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2736	. . . . . . . . . 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 41466	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊})
7	6	eleq2d 2822	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊}))
8		breq1 5101	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → (𝑝(le‘𝐾)𝑊 ↔ 𝑋(le‘𝐾)𝑊))
9	8	notbid 318	. . . . . . . . 9 ⊢ (𝑝 = 𝑋 → (¬ 𝑝(le‘𝐾)𝑊 ↔ ¬ 𝑋(le‘𝐾)𝑊))
10	9	elrab 3646	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑝 ∈ (Atoms‘𝐾) ∣ ¬ 𝑝(le‘𝐾)𝑊} ↔ (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊))
11	7, 10	bitrdi 287	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊)))
12	11	biimpa 476	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋 ∈ (Atoms‘𝐾) ∧ ¬ 𝑋(le‘𝐾)𝑊))
13		eqid 2736	. . . . . . 7 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
14		eqid 2736	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
15		eqid 2736	. . . . . . 7 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
16	2, 3, 4, 13, 14, 15, 5	dicval 41458	. . . . . 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 591	. . . . 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 5728	. . . 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 258	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → Rel (𝐼‘𝑋))
20	19	ex 412	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 → Rel (𝐼‘𝑋)))
21		rel0 5748	. . 3 ⊢ Rel ∅
22		ndmfv 6866	. . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (𝐼‘𝑋) = ∅)
23	22	releqd 5728	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐼 → (Rel (𝐼‘𝑋) ↔ Rel ∅))
24	21, 23	mpbiri 258	. 2 ⊢ (¬ 𝑋 ∈ dom 𝐼 → Rel (𝐼‘𝑋))
25	20, 24	pm2.61d1 180	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3399  ∅c0 4285   class class class wbr 5098  {copab 5160  dom cdm 5624  Rel wrel 5629  ‘cfv 6492  ℩crio 7314  lecple 17186  occoc 17187  Atomscatm 39545  LHypclh 40266  LTrncltrn 40383  TEndoctendo 41034  DIsoCcdic 41454

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-dic 41455

This theorem is referenced by: (None)