Theorem iserd 6499

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 2158	. . 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 1854	. . . . 5 |- ( ph -> A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
9	8	alrimiv 1854	. . . 4 |- ( ph -> A. y A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
10	9	alrimiv 1854	. . 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 6474	. . 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 1166	. 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 6493	. . . . . . 7 |- ( ( ph /\ x e. dom R ) -> x R x )
16	15	ex 114	. . . . . 6 |- ( ph -> ( x e. dom R -> x R x ) )
17		vex 2715	. . . . . . 7 |- x e. _V
18	17, 17	breldm 4787	. . . . . 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 2155	. . 3 |- ( ph -> dom R = A )
23		ereq2 6481	. . 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 1333   = wceq 1335   e. wcel 2128   class class class wbr 3965   dom cdm 4583   Rel wrel 4588   Er wer 6470

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-er 6473

This theorem is referenced by:  swoer  6501  eqer  6505  0er  6507  iinerm  6545  erinxp  6547  ecopover  6571  ecopoverg  6574  ener  6717  enq0er  7338  xmeter  12796