Theorem pats 40533

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 3475	. 2 ⊢ (𝐾 ∈ 𝐷 → 𝐾 ∈ V)
2		patoms.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		fveq2 6905	. . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4		patoms.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2816	. . . . 5 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6		fveq2 6905	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = ( ⋖ ‘𝐾))
7		patoms.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
8	6, 7	eqtr4di 2816	. . . . . . 7 ⊢ (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = 𝐶)
9	8	breqd 5117	. . . . . 6 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ (0.‘𝑝)𝐶𝑥))
10		fveq2 6905	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
11		patoms.z	. . . . . . . 8 ⊢ 0 = (0.‘𝐾)
12	10, 11	eqtr4di 2816	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
13	12	breq1d 5116	. . . . . 6 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)𝐶𝑥 ↔ 0 𝐶𝑥))
14	9, 13	bitrd 282	. . . . 5 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ 0 𝐶𝑥))
15	5, 14	rabeqbidv 3433	. . . 4 ⊢ (𝑝 = 𝐾 → {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥} = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
16		df-ats 40515	. . . 4 ⊢ Atoms = (𝑝 ∈ V ↦ {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥})
17	4	fvexi 6919	. . . . 5 ⊢ 𝐵 ∈ V
18	17	rabex 5304	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥} ∈ V
19	15, 16, 18	fvmpt 7013	. . 3 ⊢ (𝐾 ∈ V → (Atoms‘𝐾) = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
20	2, 19	eqtrid 2810	. 2 ⊢ (𝐾 ∈ V → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
21	1, 20	syl 18	1 ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454   class class class wbr 5106  ‘cfv 6555  Basecbs 17413  0.cp0 18621   ⋖ ccvr 40510  Atomscatm 40511

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6509  df-fun 6557  df-fv 6563  df-ats 40515

This theorem is used by:  isat  40534