Theorem pats 37293

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 3449	. 2 ⊢ (𝐾 ∈ 𝐷 → 𝐾 ∈ V)
2		patoms.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		fveq2 6769	. . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4		patoms.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2798	. . . . 5 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6		fveq2 6769	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = ( ⋖ ‘𝐾))
7		patoms.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
8	6, 7	eqtr4di 2798	. . . . . . 7 ⊢ (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = 𝐶)
9	8	breqd 5090	. . . . . 6 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ (0.‘𝑝)𝐶𝑥))
10		fveq2 6769	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
11		patoms.z	. . . . . . . 8 ⊢ 0 = (0.‘𝐾)
12	10, 11	eqtr4di 2798	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
13	12	breq1d 5089	. . . . . 6 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)𝐶𝑥 ↔ 0 𝐶𝑥))
14	9, 13	bitrd 278	. . . . 5 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ 0 𝐶𝑥))
15	5, 14	rabeqbidv 3419	. . . 4 ⊢ (𝑝 = 𝐾 → {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥} = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
16		df-ats 37275	. . . 4 ⊢ Atoms = (𝑝 ∈ V ↦ {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥})
17	4	fvexi 6783	. . . . 5 ⊢ 𝐵 ∈ V
18	17	rabex 5260	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥} ∈ V
19	15, 16, 18	fvmpt 6870	. . 3 ⊢ (𝐾 ∈ V → (Atoms‘𝐾) = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
20	2, 19	eqtrid 2792	. 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 1542   ∈ wcel 2110  {crab 3070  Vcvv 3431   class class class wbr 5079  ‘cfv 6431  Basecbs 16908  0.cp0 18137   ⋖ ccvr 37270  Atomscatm 37271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6389  df-fun 6433  df-fv 6439  df-ats 37275

This theorem is referenced by:  isat  37294