Theorem iserd 6132

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 2041	. . 3 |- ( ph -> dom R = dom R )
3		iserd.2	. . . . . . . 8 |- ( ( ph /\ x R y ) -> y R x )
4	3	ex 108	. . . . . . 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 108	. . . . . . 7 |- ( ph -> ( ( x R y /\ y R z ) -> x R z ) )
7	4, 6	jca 290	. . . . . 6 |- ( ph -> ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
8	7	alrimiv 1754	. . . . 5 |- ( ph -> A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
9	8	alrimiv 1754	. . . 4 |- ( ph -> A. y A. z ( ( x R y -> y R x ) /\ ( ( x R y /\ y R z ) -> x R z ) ) )
10	9	alrimiv 1754	. . 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 6107	. . 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 1088	. 2 |- ( ph -> R Er dom R )
13	12	adantr 261	. . . . . . . 8 |- ( ( ph /\ x e. dom R ) -> R Er dom R )
14		simpr 103	. . . . . . . 8 |- ( ( ph /\ x e. dom R ) -> x e. dom R )
15	13, 14	erref 6126	. . . . . . 7 |- ( ( ph /\ x e. dom R ) -> x R x )
16	15	ex 108	. . . . . 6 |- ( ph -> ( x e. dom R -> x R x ) )
17		vex 2560	. . . . . . 7 |- x e. _V
18	17, 17	breldm 4539	. . . . . 6 |- ( x R x -> x e. dom R )
19	16, 18	impbid1 130	. . . . 5 |- ( ph -> ( x e. dom R <-> x R x ) )
20		iserd.4	. . . . 5 |- ( ph -> ( x e. A <-> x R x ) )
21	19, 20	bitr4d 180	. . . 4 |- ( ph -> ( x e. dom R <-> x e. A ) )
22	21	eqrdv 2038	. . 3 |- ( ph -> dom R = A )
23		ereq2 6114	. . 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 135	1 |- ( ph -> R Er A )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   class class class wbr 3764   dom cdm 4345   Rel wrel 4350   Er wer 6103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-er 6106

This theorem is referenced by:  swoer  6134  eqer  6138  0er  6140  iinerm  6178  erinxp  6180  ecopover  6204  ecopoverg  6207  ener  6259  enq0er  6533