Theorem ereq1 6687

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 4801	. . 3 |- ( R = S -> ( Rel R <-> Rel S ) )
2		dmeq 4923	. . . 4 |- ( R = S -> dom R = dom S )
3	2	eqeq1d 2238	. . 3 |- ( R = S -> ( dom R = A <-> dom S = A ) )
4		cnveq 4896	. . . . . 6 |- ( R = S -> `' R = `' S )
5		coeq1 4879	. . . . . . 7 |- ( R = S -> ( R o. R ) = ( S o. R ) )
6		coeq2 4880	. . . . . . 7 |- ( R = S -> ( S o. R ) = ( S o. S ) )
7	5, 6	eqtrd 2262	. . . . . 6 |- ( R = S -> ( R o. R ) = ( S o. S ) )
8	4, 7	uneq12d 3359	. . . . 5 |- ( R = S -> ( `' R u. ( R o. R ) ) = ( `' S u. ( S o. S ) ) )
9	8	sseq1d 3253	. . . 4 |- ( R = S -> ( ( `' R u. ( R o. R ) ) C_ R <-> ( `' S u. ( S o. S ) ) C_ R ) )
10		sseq2 3248	. . . 4 |- ( R = S -> ( ( `' S u. ( S o. S ) ) C_ R <-> ( `' S u. ( S o. S ) ) C_ S ) )
11	9, 10	bitrd 188	. . 3 |- ( R = S -> ( ( `' R u. ( R o. R ) ) C_ R <-> ( `' S u. ( S o. S ) ) C_ S ) )
12	1, 3, 11	3anbi123d 1346	. 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 6680	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
14		df-er 6680	. 2 |- ( S Er A <-> ( Rel S /\ dom S = A /\ ( `' S u. ( S o. S ) ) C_ S ) )
15	12, 13, 14	3bitr4g 223	1 |- ( R = S -> ( R Er A <-> S Er A ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   = wceq 1395   u. cun 3195   C_ wss 3197   `'ccnv 4718   dom cdm 4719   o. ccom 4723   Rel wrel 4724   Er wer 6677

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-er 6680

This theorem is referenced by:  riinerm  6755