Theorem isdilN 40141

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 40140	. . 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 6839	. . . . . 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 3156	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝐹‘𝑥) = 𝑥)))
12	11	elrab 3656	. 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 3402   ⊆ wss 3911  ‘cfv 6499  Atomscatm 39249  PSubSpcpsubsp 39483  WAtomscwpointsN 39973  PAutcpautN 39974  DilcdilN 40089

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 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-dilN 40093

This theorem is referenced by: (None)