Theorem eqer 6624

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 4791	. . . 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 2202	. . . . 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 6623	. . . . 5 |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A )
8	6, 1	eqerlem 6623	. . . . 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 2214	. . . . 5 |- ( ( [_ z / x ]_ A = [_ w / x ]_ A /\ [_ w / x ]_ A = [_ v / x ]_ A ) -> [_ z / x ]_ A = [_ v / x ]_ A )
12	6, 1	eqerlem 6623	. . . . . 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 6623	. . . . 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 2766	. . . . 5 |- z e. _V
18		eqid 2196	. . . . . 6 |- [_ z / x ]_ A = [_ z / x ]_ A
19	6, 1	eqerlem 6623	. . . . . 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 6618	. 2 |- ( T. -> R Er _V )
24	23	mptru 1373	1 |- R Er _V

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   T. wtru 1365   e. wcel 2167   _Vcvv 2763   [_csb 3084   class class class wbr 4033   {copab 4093   Rel wrel 4668   Er wer 6589

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-er 6592

This theorem is referenced by:  ider  6625