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

Theorem rneq 4724
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 4671 . . 3 (𝐴 = 𝐵𝐴 = 𝐵)
21dmeqd 4699 . 2 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3 df-rn 4508 . 2 ran 𝐴 = dom 𝐴
4 df-rn 4508 . 2 ran 𝐵 = dom 𝐵
52, 3, 43eqtr4g 2170 1 (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1312  ccnv 4496  dom cdm 4497  ran crn 4498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-cnv 4505  df-dm 4507  df-rn 4508
This theorem is referenced by:  rneqi  4725  rneqd  4726  xpima1  4941  feq1  5211  foeq1  5297  ixpsnf1o  6582
  Copyright terms: Public domain W3C validator