Theorem dibffval 41119

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 3459	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6826	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		dibval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2782	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6826	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (DIsoA‘𝑘) = (DIsoA‘𝐾))
6	5	fveq1d 6828	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((DIsoA‘𝑘)‘𝑤) = ((DIsoA‘𝐾)‘𝑤))
7	6	dmeqd 5852	. . . . 5 ⊢ (𝑘 = 𝐾 → dom ((DIsoA‘𝑘)‘𝑤) = dom ((DIsoA‘𝐾)‘𝑤))
8	6	fveq1d 6828	. . . . . 6 ⊢ (𝑘 = 𝐾 → (((DIsoA‘𝑘)‘𝑤)‘𝑥) = (((DIsoA‘𝐾)‘𝑤)‘𝑥))
9		fveq2 6826	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
10	9	fveq1d 6828	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
11		fveq2 6826	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
12		dibval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
13	11, 12	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
14	13	reseq2d 5934	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( I ↾ (Base‘𝑘)) = ( I ↾ 𝐵))
15	10, 14	mpteq12dv 5182	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)))
16	15	sneqd 4591	. . . . . 6 ⊢ (𝑘 = 𝐾 → {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))} = {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})
17	8, 16	xpeq12d 5654	. . . . 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 5182	. . . 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 5182	. . 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 41118	. . 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 7168	. 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 1540   ∈ wcel 2109  Vcvv 3438  {csn 4579   ↦ cmpt 5176   I cid 5517   × cxp 5621  dom cdm 5623   ↾ cres 5625  ‘cfv 6486  Basecbs 17138  LHypclh 39963  LTrncltrn 40080  DIsoAcdia 41007  DIsoBcdib 41117

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 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-dib 41118

This theorem is referenced by:  dibfval  41120