Theorem ereq1 6596

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 4742	. . 3 |- ( R = S -> ( Rel R <-> Rel S ) )
2		dmeq 4863	. . . 4 |- ( R = S -> dom R = dom S )
3	2	eqeq1d 2202	. . 3 |- ( R = S -> ( dom R = A <-> dom S = A ) )
4		cnveq 4837	. . . . . 6 |- ( R = S -> `' R = `' S )
5		coeq1 4820	. . . . . . 7 |- ( R = S -> ( R o. R ) = ( S o. R ) )
6		coeq2 4821	. . . . . . 7 |- ( R = S -> ( S o. R ) = ( S o. S ) )
7	5, 6	eqtrd 2226	. . . . . 6 |- ( R = S -> ( R o. R ) = ( S o. S ) )
8	4, 7	uneq12d 3315	. . . . 5 |- ( R = S -> ( `' R u. ( R o. R ) ) = ( `' S u. ( S o. S ) ) )
9	8	sseq1d 3209	. . . 4 |- ( R = S -> ( ( `' R u. ( R o. R ) ) C_ R <-> ( `' S u. ( S o. S ) ) C_ R ) )
10		sseq2 3204	. . . 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 1323	. 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 6589	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
14		df-er 6589	. 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 980   = wceq 1364   u. cun 3152   C_ wss 3154   `'ccnv 4659   dom cdm 4660   o. ccom 4664   Rel wrel 4665   Er wer 6586

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-er 6589

This theorem is referenced by:  riinerm  6664