Theorem iserd 6527

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 2166	. . 3 |- ( ph -> dom R = dom R )
3		iserd.2	. . . . . . . 8 |- ( ( ph /\ x R y ) -> y R x )
4	3	ex 114	. . . . . . 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 114	. . . . . . 7 |- ( ph -> ( ( x R y /\ y R z ) -> x R z ) )
7	4, 6	jca 304	. . . . . 6 |- ( ph -> ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
8	7	alrimiv 1862	. . . . 5 |- ( ph -> A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
9	8	alrimiv 1862	. . . 4 |- ( ph -> A. y A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
10	9	alrimiv 1862	. . 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 6502	. . 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 1171	. 2 |- ( ph -> R Er dom R )
13	12	adantr 274	. . . . . . . 8 |- ( ( ph /\ x e. dom R ) -> R Er dom R )
14		simpr 109	. . . . . . . 8 |- ( ( ph /\ x e. dom R ) -> x e. dom R )
15	13, 14	erref 6521	. . . . . . 7 |- ( ( ph /\ x e. dom R ) -> x R x )
16	15	ex 114	. . . . . 6 |- ( ph -> ( x e. dom R -> x R x ) )
17		vex 2729	. . . . . . 7 |- x e. _V
18	17, 17	breldm 4808	. . . . . 6 |- ( x R x -> x e. dom R )
19	16, 18	impbid1 141	. . . . 5 |- ( ph -> ( x e. dom R <-> x R x ) )
20		iserd.4	. . . . 5 |- ( ph -> ( x e. A <-> x R x ) )
21	19, 20	bitr4d 190	. . . 4 |- ( ph -> ( x e. dom R <-> x e. A ) )
22	21	eqrdv 2163	. . 3 |- ( ph -> dom R = A )
23		ereq2 6509	. . 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 146	1 |- ( ph -> R Er A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   e. wcel 2136   class class class wbr 3982   dom cdm 4604   Rel wrel 4609   Er wer 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-er 6501

This theorem is referenced by:  swoer  6529  eqer  6533  0er  6535  iinerm  6573  erinxp  6575  ecopover  6599  ecopoverg  6602  ener  6745  enq0er  7376  xmeter  13076