Theorem rneqd 4728

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 4726	. 2 |- ( A = B -> ran A = ran B )
3	1, 2	syl 14	1 |- ( ph -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1314   ran crn 4500

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-cnv 4507  df-dm 4509  df-rn 4510

This theorem is referenced by:  resima2  4811  imaeq1  4834  imaeq2  4835  resiima  4855  elxp4  4984  elxp5  4985  funimacnv  5157  funimaexg  5165  fnima  5199  fnrnfv  5422  2ndvalg  5995  fo2nd  6010  f2ndres  6012  en1  6647  xpassen  6677  xpdom2  6678  sbthlemi4  6800  djudom  6930  exmidfodomrlemim  7005  seqeq1  10114  seqeq2  10115  seqeq3  10116  seq3val  10124  seqvalcd  10125  ennnfonelemex  11772  ennnfonelemf1  11776  restval  11969  restid2  11972  tgrest  12181  txvalex  12265  txval  12266  mopnval  12431