Theorem dilfsetN 40170

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 3455	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		dilset.l	. . 3 ⊢ 𝐿 = (Dil‘𝐾)
3		fveq2 6817	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		dilset.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2783	. . . . 5 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6817	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (PAut‘𝑘) = (PAut‘𝐾))
7		dilset.m	. . . . . . 7 ⊢ 𝑀 = (PAut‘𝐾)
8	6, 7	eqtr4di 2783	. . . . . 6 ⊢ (𝑘 = 𝐾 → (PAut‘𝑘) = 𝑀)
9		fveq2 6817	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
10		dilset.s	. . . . . . . 8 ⊢ 𝑆 = (PSubSp‘𝐾)
11	9, 10	eqtr4di 2783	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
12		fveq2 6817	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
13		dilset.w	. . . . . . . . . . 11 ⊢ 𝑊 = (WAtoms‘𝐾)
14	12, 13	eqtr4di 2783	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
15	14	fveq1d 6819	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊‘𝑑))
16	15	sseq2d 3965	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) ↔ 𝑥 ⊆ (𝑊‘𝑑)))
17	16	imbi1d 341	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)))
18	11, 17	raleqbidv 3310	. . . . . 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 3411	. . . . 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 5176	. . . 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 40124	. . . 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 7157	. . 3 ⊢ (𝐾 ∈ V → (Dil‘𝐾) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
23	2, 22	eqtrid 2777	. 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 1541   ∈ wcel 2110  ∀wral 3045  {crab 3393  Vcvv 3434   ⊆ wss 3900   ↦ cmpt 5170  ‘cfv 6477  Atomscatm 39281  PSubSpcpsubsp 39514  WAtomscwpointsN 40004  PAutcpautN 40005  DilcdilN 40120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-dilN 40124

This theorem is referenced by:  dilsetN  40171