 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  rneqd GIF version

Theorem rneqd 4958
 Description: Equality deduction for range. (Contributed by set.mm contributors, 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 4956 . 2 (A = B → ran A = ran B)
31, 2syl 15 1 (φ → ran A = ran B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  ran crn 4773 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rex 2620  df-br 4640  df-ima 4727  df-rn 4786 This theorem is referenced by:  resima2  5007  resiima  5012  rnxpid  5054  elxp4  5108  funimacnv  5168  fnima  5201
 Copyright terms: Public domain W3C validator