Theorem diafn 41023

Description: Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)

Hypotheses

Ref	Expression
diafn.b	⊢ 𝐵 = (Base‘𝐾)
diafn.l	⊢ ≤ = (le‘𝐾)
diafn.h	⊢ 𝐻 = (LHyp‘𝐾)
diafn.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)

Assertion

Ref	Expression
diafn	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})

Distinct variable groups:   𝑥, ≤   𝑥,𝐵   𝑥,𝐾   𝑥,𝑊

Allowed substitution hints:   𝐻(𝑥)   𝐼(𝑥)   𝑉(𝑥)

Proof of Theorem diafn

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

Step	Hyp	Ref	Expression
1		fvex 6835	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
2	1	rabex 5278	. . 3 ⊢ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦} ∈ V
3		eqid 2729	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦}) = (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦})
4	2, 3	fnmpti 6625	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦}) Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}
5		diafn.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
6		diafn.l	. . . 4 ⊢ ≤ = (le‘𝐾)
7		diafn.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
8		eqid 2729	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9		eqid 2729	. . . 4 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
10		diafn.i	. . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
11	5, 6, 7, 8, 9, 10	diafval 41020	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦}))
12	11	fneq1d 6575	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↔ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦}) Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}))
13	4, 12	mpbiri 258	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3394   class class class wbr 5092   ↦ cmpt 5173   Fn wfn 6477  ‘cfv 6482  Basecbs 17120  lecple 17168  LHypclh 39973  LTrncltrn 40090  trLctrl 40147  DIsoAcdia 41017

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 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371

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 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-disoa 41018

This theorem is referenced by:  diadm  41024  diaelrnN  41034  diaf11N  41038