Theorem rneqd 4833

Description: Equality deduction for range. (Contributed by NM, 4-Mar-2004.)

Hypothesis

Ref	Expression
rneqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
rneqd	|- ( ph -> ran A = ran B )

Proof of Theorem rneqd

Step	Hyp	Ref	Expression
1		rneqd.1	. 2 |- ( ph -> A = B )
2		rneq 4831	. 2 |- ( A = B -> ran A = ran B )
3	1, 2	syl 14	1 |- ( ph -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1343   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:  resima2  4918  imaeq1  4941  imaeq2  4942  resiima  4962  elxp4  5091  elxp5  5092  funimacnv  5264  funimaexg  5272  fnima  5306  fnrnfv  5533  2ndvalg  6111  fo2nd  6126  f2ndres  6128  en1  6765  xpassen  6796  xpdom2  6797  sbthlemi4  6925  djudom  7058  exmidfodomrlemim  7157  seqeq1  10383  seqeq2  10384  seqeq3  10385  seq3val  10393  seqvalcd  10394  ennnfonelemex  12347  ennnfonelemf1  12351  restval  12562  restid2  12565  tgrest  12809  txvalex  12894  txval  12895  mopnval  13082