Theorem dilfsetN 40781

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 3477	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		dilset.l	. . 3 ⊢ 𝐿 = (Dil‘𝐾)
3		fveq2 6869	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		dilset.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2817	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6869	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (PAut‘𝑘) = (PAut‘𝐾))
7		dilset.m	. . . . . . 7 ⊢ 𝑀 = (PAut‘𝐾)
8	6, 7	eqtr4di 2817	. . . . . 6 ⊢ (𝑘 = 𝐾 → (PAut‘𝑘) = 𝑀)
9		fveq2 6869	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
10		dilset.s	. . . . . . . 8 ⊢ 𝑆 = (PSubSp‘𝐾)
11	9, 10	eqtr4di 2817	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
12		fveq2 6869	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
13		dilset.w	. . . . . . . . . . 11 ⊢ 𝑊 = (WAtoms‘𝐾)
14	12, 13	eqtr4di 2817	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
15	14	fveq1d 6871	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊‘𝑑))
16	15	sseq2d 3970	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) ↔ 𝑥 ⊆ (𝑊‘𝑑)))
17	16	imbi1d 343	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)))
18	11, 17	raleqbidv 3338	. . . . . 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 3434	. . . . 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 5189	. . . 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 40735	. . . 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 7214	. . 3 ⊢ (𝐾 ∈ V → (Dil‘𝐾) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
23	2, 22	eqtrid 2811	. 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 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416  Vcvv 3456   ⊆ wss 3906   ↦ cmpt 5183  ‘cfv 6523  Atomscatm 39892  PSubSpcpsubsp 40125  WAtomscwpointsN 40615  PAutcpautN 40616  DilcdilN 40731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-dilN 40735

This theorem is referenced by:  dilsetN  40782