Theorem pmapfval 35042

Description: The projective map of a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)

Hypotheses

Ref	Expression
pmapfval.b	⊢ 𝐵 = (Base‘𝐾)
pmapfval.l	⊢ ≤ = (le‘𝐾)
pmapfval.a	⊢ 𝐴 = (Atoms‘𝐾)
pmapfval.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
pmapfval	⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))

Distinct variable groups:   𝐴,𝑎   𝑥,𝐵   𝑥,𝑎,𝐾

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑎)   𝐶(𝑥,𝑎)   ≤ (𝑥,𝑎)   𝑀(𝑥,𝑎)

Proof of Theorem pmapfval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V)
2		pmapfval.m	. . 3 ⊢ 𝑀 = (pmap‘𝐾)
3		fveq2 6191	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		pmapfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	syl6eqr 2674	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		fveq2 6191	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7		pmapfval.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
8	6, 7	syl6eqr 2674	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
10		pmapfval.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
11	9, 10	syl6eqr 2674	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
12	11	breqd 4664	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑎(le‘𝑘)𝑥 ↔ 𝑎 ≤ 𝑥))
13	8, 12	rabeqbidv 3195	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥} = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})
14	5, 13	mpteq12dv 4733	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))
15		df-pmap 34790	. . . 4 ⊢ pmap = (𝑘 ∈ V ↦ (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}))
16		fvex 6201	. . . . . 6 ⊢ (Base‘𝐾) ∈ V
17	4, 16	eqeltri 2697	. . . . 5 ⊢ 𝐵 ∈ V
18	17	mptex 6486	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) ∈ V
19	14, 15, 18	fvmpt 6282	. . 3 ⊢ (𝐾 ∈ V → (pmap‘𝐾) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))
20	2, 19	syl5eq 2668	. 2 ⊢ (𝐾 ∈ V → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))
21	1, 20	syl 17	1 ⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  Basecbs 15857  lecple 15948  Atomscatm 34550  pmapcpmap 34783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-pmap 34790

This theorem is referenced by:  pmapval  35043