Theorem rneq 4831

Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996.)

Assertion

Ref	Expression
rneq	|- ( A = B -> ran A = ran B )

Proof of Theorem rneq

Step	Hyp	Ref	Expression
1		cnveq 4778	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4806	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4615	. 2 |- ran A = dom `' A
4		df-rn 4615	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2224	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1343   `'ccnv 4603   dom cdm 4604   ran crn 4605

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  rneqi  4832  rneqd  4833  xpima1  5050  feq1  5320  foeq1  5406  ixpsnf1o  6702