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Theorem rneqd 4490
Description: Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
Hypothesis
Ref Expression
rneqd.1 (φA = B)
Assertion
Ref Expression
rneqd (φ → ran A = ran B)

Proof of Theorem rneqd
StepHypRef Expression
1 rneqd.1 . 2 (φA = B)
2 rneq 4488 . 2 (A = B → ran A = ran B)
31, 2syl 14 1 (φ → ran A = ran B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228  ran crn 4273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-cnv 4280  df-dm 4282  df-rn 4283
This theorem is referenced by:  resima2  4571  imaeq1  4590  imaeq2  4591  resiima  4610  elxp4  4735  elxp5  4736  funimacnv  4901  funimaexg  4909  fnima  4943  fnrnfv  5145  2ndvalg  5693  fo2nd  5708  f2ndres  5710
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