Theorem diafn 37055

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 6424	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
2	1	rabex 5007	. . 3 ⊢ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦} ∈ V
3		eqid 2799	. . 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 6233	. 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 2799	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
9		eqid 2799	. . . 4 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
10		diafn.i	. . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
11	5, 6, 7, 8, 9, 10	diafval 37052	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∣ (((trL‘𝐾)‘𝑊)‘𝑓) ≤ 𝑦}))
12	11	fneq1d 6192	. 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 250	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  {crab 3093   class class class wbr 4843   ↦ cmpt 4922   Fn wfn 6096  ‘cfv 6101  Basecbs 16184  lecple 16274  LHypclh 36005  LTrncltrn 36122  trLctrl 36179  DIsoAcdia 37049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-disoa 37050

This theorem is referenced by:  diadm  37056  diaelrnN  37066  diaf11N  37070