Theorem ereq1 6313

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 4533	. . 3 |- ( R = S -> ( Rel R <-> Rel S ) )
2		dmeq 4649	. . . 4 |- ( R = S -> dom R = dom S )
3	2	eqeq1d 2097	. . 3 |- ( R = S -> ( dom R = A <-> dom S = A ) )
4		cnveq 4623	. . . . . 6 |- ( R = S -> `' R = `' S )
5		coeq1 4606	. . . . . . 7 |- ( R = S -> ( R o. R ) = ( S o. R ) )
6		coeq2 4607	. . . . . . 7 |- ( R = S -> ( S o. R ) = ( S o. S ) )
7	5, 6	eqtrd 2121	. . . . . 6 |- ( R = S -> ( R o. R ) = ( S o. S ) )
8	4, 7	uneq12d 3156	. . . . 5 |- ( R = S -> ( `' R u. ( R o. R ) ) = ( `' S u. ( S o. S ) ) )
9	8	sseq1d 3054	. . . 4 |- ( R = S -> ( ( `' R u. ( R o. R ) ) C_ R <-> ( `' S u. ( S o. S ) ) C_ R ) )
10		sseq2 3049	. . . 4 |- ( R = S -> ( ( `' S u. ( S o. S ) ) C_ R <-> ( `' S u. ( S o. S ) ) C_ S ) )
11	9, 10	bitrd 187	. . 3 |- ( R = S -> ( ( `' R u. ( R o. R ) ) C_ R <-> ( `' S u. ( S o. S ) ) C_ S ) )
12	1, 3, 11	3anbi123d 1249	. 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 6306	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
14		df-er 6306	. 2 |- ( S Er A <-> ( Rel S /\ dom S = A /\ ( `' S u. ( S o. S ) ) C_ S ) )
15	12, 13, 14	3bitr4g 222	1 |- ( R = S -> ( R Er A <-> S Er A ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 925   = wceq 1290   u. cun 2998   C_ wss 3000   `'ccnv 4450   dom cdm 4451   o. ccom 4455   Rel wrel 4456   Er wer 6303

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-er 6306

This theorem is referenced by:  riinerm  6379