Theorem dibfna 41646

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 6840	. . . 4 ⊢ (𝐽‘𝑦) ∈ V
2		snex 5368	. . . 4 ⊢ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))} ∈ V
3	1, 2	xpex 7696	. . 3 ⊢ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) ∈ V
4		eqid 2739	. . 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 6628	. 2 ⊢ (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽
6		eqid 2739	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		dibfna.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
8		eqid 2739	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9		eqid 2739	. . . 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 41633	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})))
13	12	fneq1d 6578	. 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 259	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn dom 𝐽)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {csn 4555   ↦ cmpt 5153   I cid 5512   × cxp 5616  dom cdm 5618   ↾ cres 5620   Fn wfn 6480  ‘cfv 6485  Basecbs 17170  LHypclh 40476  LTrncltrn 40593  DIsoAcdia 41520  DIsoBcdib 41630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-dib 41631

This theorem is referenced by:  dibdiadm  41647  dibfnN  41648  dibclN  41654