Theorem dibfna 42014

Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

Hypotheses

Ref	Expression
dibfna.h	⊢ 𝐻 = (LHyp‘𝐾)
dibfna.j	⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
dibfna.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)

Assertion

Ref	Expression
dibfna	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn dom 𝐽)

Proof of Theorem dibfna

Dummy variables 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6895	. . . 4 ⊢ (𝐽‘𝑦) ∈ V
2		snex 5408	. . . 4 ⊢ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))} ∈ V
3	1, 2	xpex 7755	. . 3 ⊢ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) ∈ V
4		eqid 2762	. . 3 ⊢ (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) = (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}))
5	3, 4	fnmpti 6679	. 2 ⊢ (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽
6		eqid 2762	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		dibfna.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
8		eqid 2762	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9		eqid 2762	. . . 4 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
10		dibfna.j	. . . 4 ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
11		dibfna.i	. . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
12	6, 7, 8, 9, 10, 11	dibfval 42001	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})))
13	12	fneq1d 6629	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐼 Fn dom 𝐽 ↔ (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽))
14	5, 13	mpbiri 261	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn dom 𝐽)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {csn 4587   ↦ cmpt 5190   I cid 5553   × cxp 5657  dom cdm 5659   ↾ cres 5661   Fn wfn 6532  ‘cfv 6537  Basecbs 17305  LHypclh 40844  LTrncltrn 40961  DIsoAcdia 41888  DIsoBcdib 41998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-dib 41999

This theorem is used by:  dibdiadm  42015  dibfnN  42016  dibclN  42022