Theorem isdilN 40137

Description: The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dilset.a	⊢ 𝐴 = (Atoms‘𝐾)
dilset.s	⊢ 𝑆 = (PSubSp‘𝐾)
dilset.w	⊢ 𝑊 = (WAtoms‘𝐾)
dilset.m	⊢ 𝑀 = (PAut‘𝐾)
dilset.l	⊢ 𝐿 = (Dil‘𝐾)

Assertion

Ref	Expression
isdilN	⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝐿‘𝐷) ↔ (𝐹 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝐹‘𝑥) = 𝑥))))

Distinct variable groups:   𝑥,𝐾   𝑥,𝑆   𝑥,𝐷   𝑥,𝐹

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐿(𝑥)   𝑀(𝑥)   𝑊(𝑥)

Proof of Theorem isdilN

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dilset.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2		dilset.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
3		dilset.w	. . . 4 ⊢ 𝑊 = (WAtoms‘𝐾)
4		dilset.m	. . . 4 ⊢ 𝑀 = (PAut‘𝐾)
5		dilset.l	. . . 4 ⊢ 𝐿 = (Dil‘𝐾)
6	1, 2, 3, 4, 5	dilsetN 40136	. . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐿‘𝐷) = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})
7	6	eleq2d 2814	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝐿‘𝐷) ↔ 𝐹 ∈ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)}))
8		fveq1 6821	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
9	8	eqeq1d 2731	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) = 𝑥 ↔ (𝐹‘𝑥) = 𝑥))
10	9	imbi2d 340	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊‘𝐷) → (𝐹‘𝑥) = 𝑥)))
11	10	ralbidv 3152	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝐹‘𝑥) = 𝑥)))
12	11	elrab 3648	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)} ↔ (𝐹 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝐹‘𝑥) = 𝑥)))
13	7, 12	bitrdi 287	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝐿‘𝐷) ↔ (𝐹 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝐹‘𝑥) = 𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3394   ⊆ wss 3903  ‘cfv 6482  Atomscatm 39246  PSubSpcpsubsp 39479  WAtomscwpointsN 39969  PAutcpautN 39970  DilcdilN 40085

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-dilN 40089

This theorem is referenced by: (None)