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Theorem rneq 5218
Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
rneq  |-  ( A  =  B  ->  ran  A  =  ran  B )

Proof of Theorem rneq
StepHypRef Expression
1 cnveq 5166 . . 3  |-  ( A  =  B  ->  `' A  =  `' B
)
21dmeqd 5195 . 2  |-  ( A  =  B  ->  dom  `' A  =  dom  `' B )
3 df-rn 5000 . 2  |-  ran  A  =  dom  `' A
4 df-rn 5000 . 2  |-  ran  B  =  dom  `' B
52, 3, 43eqtr4g 2509 1  |-  ( A  =  B  ->  ran  A  =  ran  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383   `'ccnv 4988   dom cdm 4989   ran crn 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-cnv 4997  df-dm 4999  df-rn 5000
This theorem is referenced by:  rneqi  5219  rneqd  5220  feq1  5703  foeq1  5781  fnrnfv  5904  fconst5  6113  frxp  6895  tz7.44-2  7075  tz7.44-3  7076  ixpsnf1o  7511  ordtypecbv  7945  ordtypelem3  7948  dfac8alem  8413  dfac8a  8414  dfac5lem3  8509  dfac9  8519  dfac12lem1  8526  dfac12r  8529  ackbij2  8626  isfin3ds  8712  fin23lem17  8721  fin23lem29  8724  fin23lem30  8725  fin23lem32  8727  fin23lem34  8729  fin23lem35  8730  fin23lem39  8733  fin23lem41  8735  isf33lem  8749  isf34lem6  8763  dcomex  8830  axdc2lem  8831  zorn2lem1  8879  zorn2g  8886  ttukey2g  8899  gruurn  9179  rpnnen1  11224  mpfrcl  18166  ply1frcl  18334  pnrmopn  19822  isi1f  22059  itg1val  22068  axlowdimlem13  24235  axlowdim1  24240  iscusgra  24434  isuvtx  24466  wwlk  24659  clwwlk  24744  rusgra0edg  24933  isfrgra  24968  ex-rn  25139  gidval  25193  grpoinvfval  25204  grpodivfval  25222  gxfval  25237  isablo  25263  elghomlem1OLD  25341  iscom2  25392  isdivrngo  25411  vci  25419  isvclem  25448  isnvlem  25481  isphg  25710  pj11i  26607  hmopidmch  27050  hmopidmpj  27051  pjss1coi  27060  locfinreflem  27821  locfinref  27822  issibf  28253  sitgfval  28261  mrsubvrs  28860  ghomgrplem  29007  elgiso  29014  relexprn  29037  dfrtrcl2  29049  rdgprc0  29202  rdgprc  29203  dfrdg2  29204  brrangeg  29562  volsupnfl  30035  dnnumch1  30966  aomclem3  30978  aomclem8  30983  csbima12gALTVD  33565
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