Theorem eqer 6304

Description: Equivalence relation involving equality of dependent classes A ( x ) and B ( y ). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
eqer.1	|- ( x = y -> A = B )
eqer.2	|- R = { <. x , y >. | A = B }

Assertion

Ref	Expression
eqer	|- R Er _V

Distinct variable groups:   x, y   y, A   x, B

Allowed substitution hints:   A( x)   B( y)   R( x, y)

Proof of Theorem eqer

Dummy variables w z v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqer.2	. . . . 5 |- R = { <. x , y >. | A = B }
2	1	relopabi 4551	. . . 4 |- Rel R
3	2	a1i 9	. . 3 |- ( T. -> Rel R )
4		id 19	. . . . . 6 |- ( [_ z / x ]_ A = [_ w / x ]_ A -> [_ z / x ]_ A = [_ w / x ]_ A )
5	4	eqcomd 2093	. . . . 5 |- ( [_ z / x ]_ A = [_ w / x ]_ A -> [_ w / x ]_ A = [_ z / x ]_ A )
6		eqer.1	. . . . . 6 |- ( x = y -> A = B )
7	6, 1	eqerlem 6303	. . . . 5 |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A )
8	6, 1	eqerlem 6303	. . . . 5 |- ( w R z <-> [_ w / x ]_ A = [_ z / x ]_ A )
9	5, 7, 8	3imtr4i 199	. . . 4 |- ( z R w -> w R z )
10	9	adantl 271	. . 3 |- ( ( T. /\ z R w ) -> w R z )
11		eqtr 2105	. . . . 5 |- ( ( [_ z / x ]_ A = [_ w / x ]_ A /\ [_ w / x ]_ A = [_ v / x ]_ A ) -> [_ z / x ]_ A = [_ v / x ]_ A )
12	6, 1	eqerlem 6303	. . . . . 6 |- ( w R v <-> [_ w / x ]_ A = [_ v / x ]_ A )
13	7, 12	anbi12i 448	. . . . 5 |- ( ( z R w /\ w R v ) <-> ( [_ z / x ]_ A = [_ w / x ]_ A /\ [_ w / x ]_ A = [_ v / x ]_ A ) )
14	6, 1	eqerlem 6303	. . . . 5 |- ( z R v <-> [_ z / x ]_ A = [_ v / x ]_ A )
15	11, 13, 14	3imtr4i 199	. . . 4 |- ( ( z R w /\ w R v ) -> z R v )
16	15	adantl 271	. . 3 |- ( ( T. /\ ( z R w /\ w R v ) ) -> z R v )
17		vex 2622	. . . . 5 |- z e. _V
18		eqid 2088	. . . . . 6 |- [_ z / x ]_ A = [_ z / x ]_ A
19	6, 1	eqerlem 6303	. . . . . 6 |- ( z R z <-> [_ z / x ]_ A = [_ z / x ]_ A )
20	18, 19	mpbir 144	. . . . 5 |- z R z
21	17, 20	2th 172	. . . 4 |- ( z e. _V <-> z R z )
22	21	a1i 9	. . 3 |- ( T. -> ( z e. _V <-> z R z ) )
23	3, 10, 16, 22	iserd 6298	. 2 |- ( T. -> R Er _V )
24	23	mptru 1298	1 |- R Er _V

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   T. wtru 1290   e. wcel 1438   _Vcvv 2619   [_csb 2931   class class class wbr 3837   {copab 3890   Rel wrel 4433   Er wer 6269

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-er 6272

This theorem is referenced by:  ider  6305