Theorem iserd 6823

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 2239	. . 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 1927	. . . . 5 |- ( ph -> A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
9	8	alrimiv 1927	. . . 4 |- ( ph -> A. y A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
10	9	alrimiv 1927	. . 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 6798	. . 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 1212	. 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 6817	. . . . . . 7 |- ( ( ph /\ x e. dom R ) -> x R x )
16	15	ex 115	. . . . . 6 |- ( ph -> ( x e. dom R -> x R x ) )
17		vex 2824	. . . . . . 7 |- x e. _V
18	17, 17	breldm 4980	. . . . . 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 2236	. . 3 |- ( ph -> dom R = A )
23		ereq2 6805	. . 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 1400   = wceq 1402   e. wcel 2209   class class class wbr 4125   dom cdm 4769   Rel wrel 4774   Er wer 6794

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-er 6797

This theorem is referenced by:  swoer  6825  eqer  6829  0er  6831  iinerm  6871  erinxp  6873  ecopover  6897  ecopoverg  6900  ener  7056  enq0er  7792  eqger  14004  xmeter  15460