Theorem dicfnN 41985

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 5111	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑝 ≤ 𝑊 ↔ 𝑞 ≤ 𝑊))
2	1	notbid 321	. . . . . 6 ⊢ (𝑝 = 𝑞 → (¬ 𝑝 ≤ 𝑊 ↔ ¬ 𝑞 ≤ 𝑊))
3	2	elrab 3649	. . . . 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 2762	. . . . . . 7 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
8		eqid 2762	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9		eqid 2762	. . . . . . 7 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
10		dicfn.i	. . . . . . 7 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
11	4, 5, 6, 7, 8, 9, 10	dicval 41978	. . . . . 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 6894	. . . . . 6 ⊢ (𝐼‘𝑞) ∈ V
13	11, 12	eqeltrrdi 2871	. . . . 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 605	. . . 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 3156	. . 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 2762	. . . 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 6675	. . 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 18	. 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 41977	. . 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 6628	. 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 260	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  {crab 3415  Vcvv 3454   class class class wbr 5108  {copab 5172   ↦ cmpt 5191   Fn wfn 6531  ‘cfv 6536  ℩crio 7368  lecple 17323  occoc 17324  Atomscatm 40065  LHypclh 40786  LTrncltrn 40903  TEndoctendo 41554  DIsoCcdic 41974

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-dic 41975

This theorem is used by:  dicdmN  41986