Theorem dicfnN 38318

Description: Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dicfn.l	⊢ ≤ = (le‘𝐾)
dicfn.a	⊢ 𝐴 = (Atoms‘𝐾)
dicfn.h	⊢ 𝐻 = (LHyp‘𝐾)
dicfn.i	⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)

Assertion

Ref	Expression
dicfnN	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊})

Distinct variable groups:   ≤ ,𝑝   𝐴,𝑝   𝐾,𝑝   𝑊,𝑝

Allowed substitution hints:   𝐻(𝑝)   𝐼(𝑝)   𝑉(𝑝)

Proof of Theorem dicfnN

Dummy variables 𝑞 𝑓 𝑠 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5068	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑝 ≤ 𝑊 ↔ 𝑞 ≤ 𝑊))
2	1	notbid 320	. . . . . 6 ⊢ (𝑝 = 𝑞 → (¬ 𝑝 ≤ 𝑊 ↔ ¬ 𝑞 ≤ 𝑊))
3	2	elrab 3679	. . . . 5 ⊢ (𝑞 ∈ {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊} ↔ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊))
4		dicfn.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
5		dicfn.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
6		dicfn.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
7		eqid 2821	. . . . . . 7 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
8		eqid 2821	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9		eqid 2821	. . . . . . 7 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
10		dicfn.i	. . . . . . 7 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
11	4, 5, 6, 7, 8, 9, 10	dicval 38311	. . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝐼‘𝑞) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
12		fvex 6682	. . . . . 6 ⊢ (𝐼‘𝑞) ∈ V
13	11, 12	eqeltrrdi 2922	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))} ∈ V)
14	3, 13	sylan2b 595	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑞 ∈ {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊}) → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))} ∈ V)
15	14	ralrimiva 3182	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ∀𝑞 ∈ {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊} {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))} ∈ V)
16		eqid 2821	. . . 4 ⊢ (𝑞 ∈ {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}) = (𝑞 ∈ {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
17	16	fnmpt 6487	. . 3 ⊢ (∀𝑞 ∈ {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊} {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))} ∈ V → (𝑞 ∈ {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}) Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊})
18	15, 17	syl 17	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑞 ∈ {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}) Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊})
19	4, 5, 6, 7, 8, 9, 10	dicfval 38310	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑞 ∈ {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}))
20	19	fneq1d 6445	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐼 Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊} ↔ (𝑞 ∈ {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑢 ∈ ((LTrn‘𝐾)‘𝑊)(𝑢‘((oc‘𝐾)‘𝑊)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}) Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊}))
21	18, 20	mpbird 259	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142  Vcvv 3494   class class class wbr 5065  {copab 5127   ↦ cmpt 5145   Fn wfn 6349  ‘cfv 6354  ℩crio 7112  lecple 16571  occoc 16572  Atomscatm 36398  LHypclh 37119  LTrncltrn 37236  TEndoctendo 37887  DIsoCcdic 38307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-dic 38308

This theorem is referenced by:  dicdmN  38319