Theorem dibfna 41600

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 6853	. . . 4 ⊢ (𝐽‘𝑦) ∈ V
2		snex 5381	. . . 4 ⊢ {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))} ∈ V
3	1, 2	xpex 7707	. . 3 ⊢ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) ∈ V
4		eqid 2736	. . 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 6641	. 2 ⊢ (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) Fn dom 𝐽
6		eqid 2736	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		dibfna.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
8		eqid 2736	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9		eqid 2736	. . . 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 41587	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑦 ∈ dom 𝐽 ↦ ((𝐽‘𝑦) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})))
13	12	fneq1d 6591	. 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 258	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn dom 𝐽)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4567   ↦ cmpt 5166   I cid 5525   × cxp 5629  dom cdm 5631   ↾ cres 5633   Fn wfn 6493  ‘cfv 6498  Basecbs 17179  LHypclh 40430  LTrncltrn 40547  DIsoAcdia 41474  DIsoBcdib 41584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-dib 41585

This theorem is referenced by:  dibdiadm  41601  dibfnN  41602  dibclN  41608