Theorem rneqd 4953

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 4951	. 2 |- ( A = B -> ran A = ran B )
3	1, 2	syl 14	1 |- ( ph -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1395   ran crn 4720

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by:  resima2  5039  imaeq1  5063  imaeq2  5064  mptimass  5081  resiima  5086  elxp4  5216  elxp5  5217  funimacnv  5397  funimaexg  5405  fnima  5442  fnrnfv  5680  2ndvalg  6289  fo2nd  6304  f2ndres  6306  en1  6951  xpassen  6989  xpdom2  6990  sbthlemi4  7127  djudom  7260  exmidfodomrlemim  7379  seqeq1  10672  seqeq2  10673  seqeq3  10674  seq3val  10682  seqvalcd  10683  s1rn  11151  ennnfonelemex  12985  ennnfonelemf1  12989  restval  13278  restid2  13281  prdsex  13302  prdsval  13306  imasival  13339  conjsubg  13814  rnrhmsubrg  14216  tgrest  14843  txvalex  14928  txval  14929  mopnval  15116  edgvalg  15860  edgopval  15862  edgstruct  15864  uhgr2edg  16004