Theorem isat 34573

Description: The predicate "is an atom". (ela 29198 analog.) (Contributed by NM, 18-Sep-2011.)

Hypotheses

Ref	Expression
isatom.b	|- B = ( Base ` K )
isatom.z	|- .0. = ( 0. ` K )
isatom.c	|- C = ( <o ` K )
isatom.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
isat	|- ( K e. D -> ( P e. A <-> ( P e. B /\ .0. C P ) ) )

Proof of Theorem isat

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isatom.b	. . . 4 |- B = ( Base ` K )
2		isatom.z	. . . 4 |- .0. = ( 0. ` K )
3		isatom.c	. . . 4 |- C = ( <o ` K )
4		isatom.a	. . . 4 |- A = ( Atoms ` K )
5	1, 2, 3, 4	pats 34572	. . 3 |- ( K e. D -> A = { x e. B | .0. C x } )
6	5	eleq2d 2687	. 2 |- ( K e. D -> ( P e. A <-> P e. { x e. B | .0. C x } ) )
7		breq2 4657	. . 3 |- ( x = P -> ( .0. C x <-> .0. C P ) )
8	7	elrab 3363	. 2 |- ( P e. { x e. B | .0. C x } <-> ( P e. B /\ .0. C P ) )
9	6, 8	syl6bb 276	1 |- ( K e. D -> ( P e. A <-> ( P e. B /\ .0. C P ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653   ` cfv 5888   Basecbs 15857   0.cp0 17037   <o ccvr 34549   Atomscatm 34550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ats 34554

This theorem is referenced by:  isat2  34574  atcvr0  34575  atbase  34576  isat3  34594  1cvrco  34758  1cvrjat  34761  ltrnatb  35423