ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rneqd Structured version   GIF version

Theorem rneqd 4506
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 4504 . 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 1242  ran crn 4289
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  resima2  4587  imaeq1  4606  imaeq2  4607  resiima  4626  elxp4  4751  elxp5  4752  funimacnv  4918  funimaexg  4926  fnima  4960  fnrnfv  5163  2ndvalg  5712  fo2nd  5727  f2ndres  5729  en1  6215  xpassen  6240  xpdom2  6241  iseqeq1  8854  iseqeq2  8855  iseqeq3  8856  iseqeq4  8857  iseqval  8860
  Copyright terms: Public domain W3C validator