Theorem isldil 40917

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 40916	. . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)})
7	6	eleq2d 2851	. 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ 𝐹 ∈ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)}))
8		fveq1 6884	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
9	8	eqeq1d 2767	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) = 𝑥 ↔ (𝐹‘𝑥) = 𝑥))
10	9	imbi2d 343	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))
11	10	ralbidv 3190	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))
12	11	elrab 3652	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ 𝐼 ∣ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝑓‘𝑥) = 𝑥)} ↔ (𝐹 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))
13	7, 12	bitrdi 290	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418   class class class wbr 5111  ‘cfv 6540  Basecbs 17287  lecple 17335  LHypclh 40791  LAutclaut 40792  LDilcldil 40907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ldil 40911

This theorem is used by:  ldillaut  40918  ldilval  40920  idldil  40921  ldilcnv  40922  ldilco  40923  cdleme50ldil  41355