Theorem isldil 39447

Description: The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Hypotheses

Ref	Expression
ldilset.b	⊢ 𝐵 = (Base‘𝐾)
ldilset.l	⊢ ≤ = (le‘𝐾)
ldilset.h	⊢ 𝐻 = (LHyp‘𝐾)
ldilset.i	⊢ 𝐼 = (LAut‘𝐾)
ldilset.d	⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)

Assertion

Ref	Expression
isldil	⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))))

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

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)   𝐻(𝑥)   𝐼(𝑥)   ≤ (𝑥)

Proof of Theorem isldil

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ldilset.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		ldilset.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		ldilset.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4		ldilset.i	. . . 4 ⊢ 𝐼 = (LAut‘𝐾)
5		ldilset.d	. . . 4 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	ldilset 39446	. . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})
7	6	eleq2d 2818	. 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ 𝐹 ∈ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)}))
8		fveq1 6890	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
9	8	eqeq1d 2733	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) = 𝑥 ↔ (𝐹‘𝑥) = 𝑥))
10	9	imbi2d 340	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))
11	10	ralbidv 3176	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))
12	11	elrab 3683	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)} ↔ (𝐹 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))
13	7, 12	bitrdi 287	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  {crab 3431   class class class wbr 5148  ‘cfv 6543  Basecbs 17151  lecple 17211  LHypclh 39321  LAutclaut 39322  LDilcldil 39437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ldil 39441

This theorem is referenced by:  ldillaut  39448  ldilval  39450  idldil  39451  ldilcnv  39452  ldilco  39453  cdleme50ldil  39885