Theorem pats 39555

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 3461	. 2 ⊢ (𝐾 ∈ 𝐷 → 𝐾 ∈ V)
2		patoms.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		fveq2 6834	. . . . . 6 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4		patoms.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2789	. . . . 5 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6		fveq2 6834	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = ( ⋖ ‘𝐾))
7		patoms.c	. . . . . . . 8 ⊢ 𝐶 = ( ⋖ ‘𝐾)
8	6, 7	eqtr4di 2789	. . . . . . 7 ⊢ (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = 𝐶)
9	8	breqd 5109	. . . . . 6 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ (0.‘𝑝)𝐶𝑥))
10		fveq2 6834	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
11		patoms.z	. . . . . . . 8 ⊢ 0 = (0.‘𝐾)
12	10, 11	eqtr4di 2789	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
13	12	breq1d 5108	. . . . . 6 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)𝐶𝑥 ↔ 0 𝐶𝑥))
14	9, 13	bitrd 279	. . . . 5 ⊢ (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ 0 𝐶𝑥))
15	5, 14	rabeqbidv 3417	. . . 4 ⊢ (𝑝 = 𝐾 → {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥} = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
16		df-ats 39537	. . . 4 ⊢ Atoms = (𝑝 ∈ V ↦ {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥})
17	4	fvexi 6848	. . . . 5 ⊢ 𝐵 ∈ V
18	17	rabex 5284	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥} ∈ V
19	15, 16, 18	fvmpt 6941	. . 3 ⊢ (𝐾 ∈ V → (Atoms‘𝐾) = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥})
20	2, 19	eqtrid 2783	. 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 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   class class class wbr 5098  ‘cfv 6492  Basecbs 17136  0.cp0 18344   ⋖ ccvr 39532  Atomscatm 39533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ats 39537

This theorem is referenced by:  isat  39556