Theorem diafn 35341

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 6113	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
2	1	rabex 4740	. . 3 ⊢ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦} ∈ V
3		eqid 2610	. . 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 5935	. 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 2610	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9		eqid 2610	. . . 4 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
10		diafn.i	. . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
11	5, 6, 7, 8, 9, 10	diafval 35338	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦}))
12	11	fneq1d 5895	. 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 247	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   class class class wbr 4583   ↦ cmpt 4643   Fn wfn 5799  ‘cfv 5804  Basecbs 15695  lecple 15775  LHypclh 34288  LTrncltrn 34405  trLctrl 34463  DIsoAcdia 35335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-disoa 35336

This theorem is referenced by:  diadm  35342  diaelrnN  35352  diaf11N  35356