Theorem ereq1 6436

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 4621	. . 3 |- ( R = S -> ( Rel R <-> Rel S ) )
2		dmeq 4739	. . . 4 |- ( R = S -> dom R = dom S )
3	2	eqeq1d 2148	. . 3 |- ( R = S -> ( dom R = A <-> dom S = A ) )
4		cnveq 4713	. . . . . 6 |- ( R = S -> `' R = `' S )
5		coeq1 4696	. . . . . . 7 |- ( R = S -> ( R o. R ) = ( S o. R ) )
6		coeq2 4697	. . . . . . 7 |- ( R = S -> ( S o. R ) = ( S o. S ) )
7	5, 6	eqtrd 2172	. . . . . 6 |- ( R = S -> ( R o. R ) = ( S o. S ) )
8	4, 7	uneq12d 3231	. . . . 5 |- ( R = S -> ( `' R u. ( R o. R ) ) = ( `' S u. ( S o. S ) ) )
9	8	sseq1d 3126	. . . 4 |- ( R = S -> ( ( `' R u. ( R o. R ) ) C_ R <-> ( `' S u. ( S o. S ) ) C_ R ) )
10		sseq2 3121	. . . 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 1290	. 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 6429	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
14		df-er 6429	. 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 962   = wceq 1331   u. cun 3069   C_ wss 3071   `'ccnv 4538   dom cdm 4539   o. ccom 4543   Rel wrel 4544   Er wer 6426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-er 6429

This theorem is referenced by:  riinerm  6502