Theorem pats 39914

Description: The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)

Hypotheses

Ref	Expression
patoms.b	⊢ 𝐵 = (Base‘𝐾)
patoms.z	⊢ 0 = (0.‘𝐾)
patoms.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
patoms.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
pats	⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})

Distinct variable groups:   𝑥,𝐵   𝑥,𝐾

Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)   𝐷(𝑥)   0 (𝑥)

Proof of Theorem pats

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3477	. 2 ⊢ (𝐾 ∈ 𝐷 → 𝐾 ∈ V)
2		patoms.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		fveq2 6869	. . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4		patoms.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2817	. . . . 5 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6		fveq2 6869	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = ( ⋖ ‘𝐾))
7		patoms.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
8	6, 7	eqtr4di 2817	. . . . . . 7 ⊢ (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = 𝐶)
9	8	breqd 5113	. . . . . 6 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ (0.‘𝑝)𝐶𝑥))
10		fveq2 6869	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
11		patoms.z	. . . . . . . 8 ⊢ 0 = (0.‘𝐾)
12	10, 11	eqtr4di 2817	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
13	12	breq1d 5112	. . . . . 6 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)𝐶𝑥 ↔ 0 𝐶𝑥))
14	9, 13	bitrd 281	. . . . 5 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ 0 𝐶𝑥))
15	5, 14	rabeqbidv 3434	. . . 4 ⊢ (𝑝 = 𝐾 → {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥} = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
16		df-ats 39896	. . . 4 ⊢ Atoms = (𝑝 ∈ V ↦ {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥})
17	4	fvexi 6883	. . . . 5 ⊢ 𝐵 ∈ V
18	17	rabex 5297	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥} ∈ V
19	15, 16, 18	fvmpt 6977	. . 3 ⊢ (𝐾 ∈ V → (Atoms‘𝐾) = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
20	2, 19	eqtrid 2811	. 2 ⊢ (𝐾 ∈ V → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
21	1, 20	syl 17	1 ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144  {crab 3416  Vcvv 3456   class class class wbr 5102  ‘cfv 6523  Basecbs 17247  0.cp0 18455   ⋖ ccvr 39891  Atomscatm 39892

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-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-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  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-iota 6479  df-fun 6525  df-fv 6531  df-ats 39896

This theorem is referenced by:  isat  39915