Theorem dilfsetN 40954

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 3475	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		dilset.l	. . 3 ⊢ 𝐿 = (Dil‘𝐾)
3		fveq2 6881	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		dilset.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2815	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (PAut‘𝑘) = (PAut‘𝐾))
7		dilset.m	. . . . . . 7 ⊢ 𝑀 = (PAut‘𝐾)
8	6, 7	eqtr4di 2815	. . . . . 6 ⊢ (𝑘 = 𝐾 → (PAut‘𝑘) = 𝑀)
9		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
10		dilset.s	. . . . . . . 8 ⊢ 𝑆 = (PSubSp‘𝐾)
11	9, 10	eqtr4di 2815	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
12		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
13		dilset.w	. . . . . . . . . . 11 ⊢ 𝑊 = (WAtoms‘𝐾)
14	12, 13	eqtr4di 2815	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
15	14	fveq1d 6883	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊‘𝑑))
16	15	sseq2d 3968	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) ↔ 𝑥 ⊆ (𝑊‘𝑑)))
17	16	imbi1d 344	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)))
18	11, 17	raleqbidv 3337	. . . . . 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 3433	. . . . 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 5197	. . . 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 40908	. . . 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 7226	. . 3 ⊢ (𝐾 ∈ V → (Dil‘𝐾) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
23	2, 22	eqtrid 2809	. 2 ⊢ (𝐾 ∈ V → 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
24	1, 23	syl 18	1 ⊢ (𝐾 ∈ 𝐵 → 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  ∀wral 3078  {crab 3415  Vcvv 3454   ⊆ wss 3904   ↦ cmpt 5191  ‘cfv 6536  Atomscatm 40065  PSubSpcpsubsp 40298  WAtomscwpointsN 40788  PAutcpautN 40789  DilcdilN 40904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-dilN 40908

This theorem is used by:  dilsetN  40955