Theorem dibffval 40011

Description: The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)

Hypotheses

Ref	Expression
dibval.b	⊢ 𝐵 = (Base‘𝐾)
dibval.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
dibffval	⊢ (𝐾 ∈ 𝑉 → (DIsoB‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))

Distinct variable groups:   𝑤,𝐻   𝑤,𝑓,𝑥,𝐾

Allowed substitution hints:   𝐵(𝑥,𝑤,𝑓)   𝐻(𝑥,𝑓)   𝑉(𝑥,𝑤,𝑓)

Proof of Theorem dibffval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3493	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6892	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		dibval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2791	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6892	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (DIsoA‘𝑘) = (DIsoA‘𝐾))
6	5	fveq1d 6894	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((DIsoA‘𝑘)‘𝑤) = ((DIsoA‘𝐾)‘𝑤))
7	6	dmeqd 5906	. . . . 5 ⊢ (𝑘 = 𝐾 → dom ((DIsoA‘𝑘)‘𝑤) = dom ((DIsoA‘𝐾)‘𝑤))
8	6	fveq1d 6894	. . . . . 6 ⊢ (𝑘 = 𝐾 → (((DIsoA‘𝑘)‘𝑤)‘𝑥) = (((DIsoA‘𝐾)‘𝑤)‘𝑥))
9		fveq2 6892	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
10	9	fveq1d 6894	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
11		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
12		dibval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
13	11, 12	eqtr4di 2791	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
14	13	reseq2d 5982	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( I ↾ (Base‘𝑘)) = ( I ↾ 𝐵))
15	10, 14	mpteq12dv 5240	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)))
16	15	sneqd 4641	. . . . . 6 ⊢ (𝑘 = 𝐾 → {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))} = {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})
17	8, 16	xpeq12d 5708	. . . . 5 ⊢ (𝑘 = 𝐾 → ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}) = ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))
18	7, 17	mpteq12dv 5240	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))})) = (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))
19	4, 18	mpteq12dv 5240	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
20		df-dib 40010	. . 3 ⊢ DIsoB = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))))
21	19, 20, 3	mptfvmpt 7230	. 2 ⊢ (𝐾 ∈ V → (DIsoB‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
22	1, 21	syl 17	1 ⊢ (𝐾 ∈ 𝑉 → (DIsoB‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475  {csn 4629   ↦ cmpt 5232   I cid 5574   × cxp 5675  dom cdm 5677   ↾ cres 5679  ‘cfv 6544  Basecbs 17144  LHypclh 38855  LTrncltrn 38972  DIsoAcdia 39899  DIsoBcdib 40009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-dib 40010

This theorem is referenced by:  dibfval  40012