Theorem pmapfval 39455

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 3482	. 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V)
2		pmapfval.m	. . 3 ⊢ 𝑀 = (pmap‘𝐾)
3		fveq2 6901	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
4		pmapfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2784	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
6		fveq2 6901	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
7		pmapfval.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
8	6, 7	eqtr4di 2784	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
9		fveq2 6901	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
10		pmapfval.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
11	9, 10	eqtr4di 2784	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
12	11	breqd 5164	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑎(le‘𝑘)𝑥 ↔ 𝑎 ≤ 𝑥))
13	8, 12	rabeqbidv 3437	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥} = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})
14	5, 13	mpteq12dv 5244	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))
15		df-pmap 39203	. . . 4 ⊢ pmap = (𝑘 ∈ V ↦ (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}))
16	14, 15, 4	mptfvmpt 7245	. . 3 ⊢ (𝐾 ∈ V → (pmap‘𝐾) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))
17	2, 16	eqtrid 2778	. 2 ⊢ (𝐾 ∈ V → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))
18	1, 17	syl 17	1 ⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  {crab 3419  Vcvv 3462   class class class wbr 5153   ↦ cmpt 5236  ‘cfv 6554  Basecbs 17213  lecple 17273  Atomscatm 38961  pmapcpmap 39196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  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 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-pmap 39203

This theorem is referenced by:  pmapval  39456