Theorem dibffval 35447

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 3185	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6103	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		dibval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	syl6eqr 2662	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (DIsoA‘𝑘) = (DIsoA‘𝐾))
6	5	fveq1d 6105	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((DIsoA‘𝑘)‘𝑤) = ((DIsoA‘𝐾)‘𝑤))
7	6	dmeqd 5248	. . . . 5 ⊢ (𝑘 = 𝐾 → dom ((DIsoA‘𝑘)‘𝑤) = dom ((DIsoA‘𝐾)‘𝑤))
8	6	fveq1d 6105	. . . . . 6 ⊢ (𝑘 = 𝐾 → (((DIsoA‘𝑘)‘𝑤)‘𝑥) = (((DIsoA‘𝐾)‘𝑤)‘𝑥))
9		fveq2 6103	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
10	9	fveq1d 6105	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
11		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
12		dibval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
13	11, 12	syl6eqr 2662	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
14	13	reseq2d 5317	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ( I ↾ (Base‘𝑘)) = ( I ↾ 𝐵))
15	10, 14	mpteq12dv 4663	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)))
16	15	sneqd 4137	. . . . . 6 ⊢ (𝑘 = 𝐾 → {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))} = {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})
17	8, 16	xpeq12d 5064	. . . . 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 4663	. . . 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 4663	. . 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 35446	. . 3 ⊢ DIsoB = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))))
21		fvex 6113	. . . . 5 ⊢ (LHyp‘𝐾) ∈ V
22	3, 21	eqeltri 2684	. . . 4 ⊢ 𝐻 ∈ V
23	22	mptex 6390	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))) ∈ V
24	19, 20, 23	fvmpt 6191	. 2 ⊢ (𝐾 ∈ V → (DIsoB‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
25	1, 24	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 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125   ↦ cmpt 4643   I cid 4948   × cxp 5036  dom cdm 5038   ↾ cres 5040  ‘cfv 5804  Basecbs 15695  LHypclh 34288  LTrncltrn 34405  DIsoAcdia 35335  DIsoBcdib 35445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-dib 35446

This theorem is referenced by:  dibfval  35448