Theorem ereq1 6650

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 4775	. . 3 |- ( R = S -> ( Rel R <-> Rel S ) )
2		dmeq 4897	. . . 4 |- ( R = S -> dom R = dom S )
3	2	eqeq1d 2216	. . 3 |- ( R = S -> ( dom R = A <-> dom S = A ) )
4		cnveq 4870	. . . . . 6 |- ( R = S -> `' R = `' S )
5		coeq1 4853	. . . . . . 7 |- ( R = S -> ( R o. R ) = ( S o. R ) )
6		coeq2 4854	. . . . . . 7 |- ( R = S -> ( S o. R ) = ( S o. S ) )
7	5, 6	eqtrd 2240	. . . . . 6 |- ( R = S -> ( R o. R ) = ( S o. S ) )
8	4, 7	uneq12d 3336	. . . . 5 |- ( R = S -> ( `' R u. ( R o. R ) ) = ( `' S u. ( S o. S ) ) )
9	8	sseq1d 3230	. . . 4 |- ( R = S -> ( ( `' R u. ( R o. R ) ) C_ R <-> ( `' S u. ( S o. S ) ) C_ R ) )
10		sseq2 3225	. . . 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 1325	. 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 6643	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
14		df-er 6643	. 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 981   = wceq 1373   u. cun 3172   C_ wss 3174   `'ccnv 4692   dom cdm 4693   o. ccom 4697   Rel wrel 4698   Er wer 6640

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-er 6643

This theorem is referenced by:  riinerm  6718