Theorem iserd 6723

Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
iserd.1	|- ( ph -> Rel R )
iserd.2	|- ( ( ph /\ x R y ) -> y R x )
iserd.3	|- ( ( ph /\ ( x R y /\ y R z ) ) -> x R z )
iserd.4	|- ( ph -> ( x e. A <-> x R x ) )

Assertion

Ref	Expression
iserd	|- ( ph -> R Er A )

Distinct variable groups:   x, y, z, R   x, A   ph, x, y, z

Allowed substitution hints:   A( y, z)

Proof of Theorem iserd

Step	Hyp	Ref	Expression
1		iserd.1	. . 3 |- ( ph -> Rel R )
2		eqidd 2230	. . 3 |- ( ph -> dom R = dom R )
3		iserd.2	. . . . . . . 8 |- ( ( ph /\ x R y ) -> y R x )
4	3	ex 115	. . . . . . 7 |- ( ph -> ( x R y -> y R x ) )
5		iserd.3	. . . . . . . 8 |- ( ( ph /\ ( x R y /\ y R z ) ) -> x R z )
6	5	ex 115	. . . . . . 7 |- ( ph -> ( ( x R y /\ y R z ) -> x R z ) )
7	4, 6	jca 306	. . . . . 6 |- ( ph -> ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
8	7	alrimiv 1920	. . . . 5 |- ( ph -> A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
9	8	alrimiv 1920	. . . 4 |- ( ph -> A. y A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
10	9	alrimiv 1920	. . 3 |- ( ph -> A. x A. y A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
11		dfer2 6698	. . 3 |- ( R Er dom R <-> ( Rel R /\ dom R = dom R /\ A. x A. y A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) ) )
12	1, 2, 10, 11	syl3anbrc 1205	. 2 |- ( ph -> R Er dom R )
13	12	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. dom R ) -> R Er dom R )
14		simpr 110	. . . . . . . 8 |- ( ( ph /\ x e. dom R ) -> x e. dom R )
15	13, 14	erref 6717	. . . . . . 7 |- ( ( ph /\ x e. dom R ) -> x R x )
16	15	ex 115	. . . . . 6 |- ( ph -> ( x e. dom R -> x R x ) )
17		vex 2803	. . . . . . 7 |- x e. _V
18	17, 17	breldm 4933	. . . . . 6 |- ( x R x -> x e. dom R )
19	16, 18	impbid1 142	. . . . 5 |- ( ph -> ( x e. dom R <-> x R x ) )
20		iserd.4	. . . . 5 |- ( ph -> ( x e. A <-> x R x ) )
21	19, 20	bitr4d 191	. . . 4 |- ( ph -> ( x e. dom R <-> x e. A ) )
22	21	eqrdv 2227	. . 3 |- ( ph -> dom R = A )
23		ereq2 6705	. . 3 |- ( dom R = A -> ( R Er dom R <-> R Er A ) )
24	22, 23	syl 14	. 2 |- ( ph -> ( R Er dom R <-> R Er A ) )
25	12, 24	mpbid 147	1 |- ( ph -> R Er A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   e. wcel 2200   class class class wbr 4086   dom cdm 4723   Rel wrel 4728   Er wer 6694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-er 6697

This theorem is referenced by:  swoer  6725  eqer  6729  0er  6731  iinerm  6771  erinxp  6773  ecopover  6797  ecopoverg  6800  ener  6948  enq0er  7645  eqger  13801  xmeter  15150