Theorem eqer 6557

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 4746	. . . 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 2181	. . . . 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 6556	. . . . 5 |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A )
8	6, 1	eqerlem 6556	. . . . 5 |- ( w R z <-> [_ w / x ]_ A = [_ z / x ]_ A )
9	5, 7, 8	3imtr4i 201	. . . 4 |- ( z R w -> w R z )
10	9	adantl 277	. . 3 |- ( ( T. /\ z R w ) -> w R z )
11		eqtr 2193	. . . . 5 |- ( ( [_ z / x ]_ A = [_ w / x ]_ A /\ [_ w / x ]_ A = [_ v / x ]_ A ) -> [_ z / x ]_ A = [_ v / x ]_ A )
12	6, 1	eqerlem 6556	. . . . . 6 |- ( w R v <-> [_ w / x ]_ A = [_ v / x ]_ A )
13	7, 12	anbi12i 460	. . . . 5 |- ( ( z R w /\ w R v ) <-> ( [_ z / x ]_ A = [_ w / x ]_ A /\ [_ w / x ]_ A = [_ v / x ]_ A ) )
14	6, 1	eqerlem 6556	. . . . 5 |- ( z R v <-> [_ z / x ]_ A = [_ v / x ]_ A )
15	11, 13, 14	3imtr4i 201	. . . 4 |- ( ( z R w /\ w R v ) -> z R v )
16	15	adantl 277	. . 3 |- ( ( T. /\ ( z R w /\ w R v ) ) -> z R v )
17		vex 2738	. . . . 5 |- z e. _V
18		eqid 2175	. . . . . 6 |- [_ z / x ]_ A = [_ z / x ]_ A
19	6, 1	eqerlem 6556	. . . . . 6 |- ( z R z <-> [_ z / x ]_ A = [_ z / x ]_ A )
20	18, 19	mpbir 146	. . . . 5 |- z R z
21	17, 20	2th 174	. . . 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 6551	. 2 |- ( T. -> R Er _V )
24	23	mptru 1362	1 |- R Er _V

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   T. wtru 1354   e. wcel 2146   _Vcvv 2735   [_csb 3055   class class class wbr 3998   {copab 4058   Rel wrel 4625   Er wer 6522

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-er 6525

This theorem is referenced by:  ider  6558