Theorem dilfsetN 39665

Description: The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-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
dilfsetN	⊢ (𝐾 ∈ 𝐵 → 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))

Distinct variable groups:   𝐴,𝑑   𝑓,𝑑,𝑥,𝐾   𝑓,𝑀   𝑥,𝑆

Allowed substitution hints:   𝐴(𝑥,𝑓)   𝐵(𝑥,𝑓,𝑑)   𝑆(𝑓,𝑑)   𝐿(𝑥,𝑓,𝑑)   𝑀(𝑥,𝑑)   𝑊(𝑥,𝑓,𝑑)

Proof of Theorem dilfsetN

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3492	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		dilset.l	. . 3 ⊢ 𝐿 = (Dil‘𝐾)
3		fveq2 6902	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		dilset.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2786	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6902	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (PAut‘𝑘) = (PAut‘𝐾))
7		dilset.m	. . . . . . 7 ⊢ 𝑀 = (PAut‘𝐾)
8	6, 7	eqtr4di 2786	. . . . . 6 ⊢ (𝑘 = 𝐾 → (PAut‘𝑘) = 𝑀)
9		fveq2 6902	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
10		dilset.s	. . . . . . . 8 ⊢ 𝑆 = (PSubSp‘𝐾)
11	9, 10	eqtr4di 2786	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
12		fveq2 6902	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
13		dilset.w	. . . . . . . . . . 11 ⊢ 𝑊 = (WAtoms‘𝐾)
14	12, 13	eqtr4di 2786	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
15	14	fveq1d 6904	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊‘𝑑))
16	15	sseq2d 4014	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) ↔ 𝑥 ⊆ (𝑊‘𝑑)))
17	16	imbi1d 340	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)))
18	11, 17	raleqbidv 3340	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)))
19	8, 18	rabeqbidv 3448	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥)} = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)})
20	5, 19	mpteq12dv 5243	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥)}) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
21		df-dilN 39619	. . . 4 ⊢ Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
22	20, 21, 4	mptfvmpt 7246	. . 3 ⊢ (𝐾 ∈ V → (Dil‘𝐾) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
23	2, 22	eqtrid 2780	. 2 ⊢ (𝐾 ∈ V → 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
24	1, 23	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∀wral 3058  {crab 3430  Vcvv 3473   ⊆ wss 3949   ↦ cmpt 5235  ‘cfv 6553  Atomscatm 38775  PSubSpcpsubsp 39009  WAtomscwpointsN 39499  PAutcpautN 39500  DilcdilN 39615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-dilN 39619

This theorem is referenced by:  dilsetN  39666