Theorem diafn 41828

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 6894	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
2	1	rabex 5309	. . 3 ⊢ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦} ∈ V
3		eqid 2763	. . 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 6678	. 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 2763	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9		eqid 2763	. . . 4 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
10		diafn.i	. . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
11	5, 6, 7, 8, 9, 10	diafval 41825	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦}))
12	11	fneq1d 6628	. 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 261	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416   class class class wbr 5109   ↦ cmpt 5192   Fn wfn 6531  ‘cfv 6536  Basecbs 17264  lecple 17312  LHypclh 40778  LTrncltrn 40895  trLctrl 40952  DIsoAcdia 41822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-disoa 41823

This theorem is referenced by:  diadm  41829  diaelrnN  41839  diaf11N  41843