Theorem ereq1 6049

Description: Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
ereq1	|- ( R = S -> ( R Er A <-> S Er A))

Proof of Theorem ereq1

Step	Hyp	Ref	Expression
1		releq 4365	. . 3 |- ( R = S -> ( Rel R <-> Rel S))
2		dmeq 4478	. . . 4 |- ( R = S -> dom R = dom S)
3	2	eqeq1d 2045	. . 3 |- ( R = S -> ( dom R = A <-> dom S = A))
4		cnveq 4452	. . . . . 6 |- ( R = S -> `' R = `' S)
5		coeq1 4436	. . . . . . 7 |- ( R = S -> ( R o. R) = ( S o. R))
6		coeq2 4437	. . . . . . 7 |- ( R = S -> ( S o. R) = ( S o. S))
7	5, 6	eqtrd 2069	. . . . . 6 |- ( R = S -> ( R o. R) = ( S o. S))
8	4, 7	uneq12d 3092	. . . . 5 |- ( R = S -> ( `' R u. ( R o. R)) = ( `' S u. ( S o. S)))
9	8	sseq1d 2966	. . . 4 |- ( R = S -> (( `' R u. ( R o. R)) C_ R <-> ( `' S u. ( S o. S)) C_ R))
10		sseq2 2961	. . . 4 |- ( R = S -> (( `' S u. ( S o. S)) C_ R <-> ( `' S u. ( S o. S)) C_ S))
11	9, 10	bitrd 177	. . 3 |- ( R = S -> (( `' R u. ( R o. R)) C_ R <-> ( `' S u. ( S o. S)) C_ S))
12	1, 3, 11	3anbi123d 1206	. 2 |- ( R = S -> (( Rel R /\ dom R = A /\ ( `' R u. ( R o. R)) C_ R) <-> ( Rel S /\ dom S = A /\ ( `' S u. ( S o. S)) C_ S)))
13		df-er 6042	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R)) C_ R))
14		df-er 6042	. 2 |- ( S Er A <-> ( Rel S /\ dom S = A /\ ( `' S u. ( S o. S)) C_ S))
15	12, 13, 14	3bitr4g 212	1 |- ( R = S -> ( R Er A <-> S Er A))

Syntax hints:   -> wi 4   <-> wb 98   /\ w3a 884   = wceq 1242   u. cun 2909   C_ wss 2911   `'ccnv 4287   dom cdm 4288   o. ccom 4292   Rel wrel 4293   Er wer 6039

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-er 6042

This theorem is referenced by:  riinerm  6115