Theorem rneqd 4763

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 4761	. 2 |- ( A = B -> ran A = ran B )
3	1, 2	syl 14	1 |- ( ph -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1331   ran crn 4535

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 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-cnv 4542  df-dm 4544  df-rn 4545

This theorem is referenced by:  resima2  4848  imaeq1  4871  imaeq2  4872  resiima  4892  elxp4  5021  elxp5  5022  funimacnv  5194  funimaexg  5202  fnima  5236  fnrnfv  5461  2ndvalg  6034  fo2nd  6049  f2ndres  6051  en1  6686  xpassen  6717  xpdom2  6718  sbthlemi4  6841  djudom  6971  exmidfodomrlemim  7050  seqeq1  10214  seqeq2  10215  seqeq3  10216  seq3val  10224  seqvalcd  10225  ennnfonelemex  11916  ennnfonelemf1  11920  restval  12115  restid2  12118  tgrest  12327  txvalex  12412  txval  12413  mopnval  12600