Theorem dibfnN 41175

Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
dibfnN	⊢ ((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 dibfnN

Step	Hyp	Ref	Expression
1		dibfn.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2735	. . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
3		dibfn.i	. . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
4	1, 2, 3	dibfna 41173	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn dom ((DIsoA‘𝐾)‘𝑊))
5		dibfn.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
6		dibfn.l	. . . 4 ⊢ ≤ = (le‘𝐾)
7	5, 6, 1, 2	diadm 41054	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom ((DIsoA‘𝐾)‘𝑊) = {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})
8	7	fneq2d 6632	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐼 Fn dom ((DIsoA‘𝐾)‘𝑊) ↔ 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}))
9	4, 8	mpbid 232	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {crab 3415   class class class wbr 5119  dom cdm 5654   Fn wfn 6526  ‘cfv 6531  Basecbs 17228  lecple 17278  LHypclh 40003  DIsoAcdia 41047  DIsoBcdib 41157

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-disoa 41048  df-dib 41158

This theorem is referenced by:  dibdmN  41176  dibf11N  41180