Theorem rneqd 4840

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 4838	. 2 |- ( A = B -> ran A = ran B )
3	1, 2	syl 14	1 |- ( ph -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1348   ran crn 4612

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by:  resima2  4925  imaeq1  4948  imaeq2  4949  resiima  4969  elxp4  5098  elxp5  5099  funimacnv  5274  funimaexg  5282  fnima  5316  fnrnfv  5543  2ndvalg  6122  fo2nd  6137  f2ndres  6139  en1  6777  xpassen  6808  xpdom2  6809  sbthlemi4  6937  djudom  7070  exmidfodomrlemim  7178  seqeq1  10404  seqeq2  10405  seqeq3  10406  seq3val  10414  seqvalcd  10415  ennnfonelemex  12369  ennnfonelemf1  12373  restval  12585  restid2  12588  tgrest  12963  txvalex  13048  txval  13049  mopnval  13236