Theorem dicfnN 41170

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 5105	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑝 ≤ 𝑊 ↔ 𝑞 ≤ 𝑊))
2	1	notbid 318	. . . . . 6 ⊢ (𝑝 = 𝑞 → (¬ 𝑝 ≤ 𝑊 ↔ ¬ 𝑞 ≤ 𝑊))
3	2	elrab 3656	. . . . 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 2729	. . . . . . 7 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
8		eqid 2729	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9		eqid 2729	. . . . . . 7 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
10		dicfn.i	. . . . . . 7 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
11	4, 5, 6, 7, 8, 9, 10	dicval 41163	. . . . . 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 6853	. . . . . 6 ⊢ (𝐼‘𝑞) ∈ V
13	11, 12	eqeltrrdi 2837	. . . . 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 594	. . . 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 3125	. . 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 2729	. . . 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 6640	. . 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 41162	. . 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 6593	. 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 257	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑝 ∈ 𝐴 ∣ ¬ 𝑝 ≤ 𝑊})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3402  Vcvv 3444   class class class wbr 5102  {copab 5164   ↦ cmpt 5183   Fn wfn 6494  ‘cfv 6499  ℩crio 7325  lecple 17203  occoc 17204  Atomscatm 39249  LHypclh 39971  LTrncltrn 40088  TEndoctendo 40739  DIsoCcdic 41159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-dic 41160

This theorem is referenced by:  dicdmN  41171