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

Theorem rneq 4836
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 4783 . . 3 (𝐴 = 𝐵𝐴 = 𝐵)
21dmeqd 4811 . 2 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3 df-rn 4620 . 2 ran 𝐴 = dom 𝐴
4 df-rn 4620 . 2 ran 𝐵 = dom 𝐵
52, 3, 43eqtr4g 2228 1 (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  ccnv 4608  dom cdm 4609  ran crn 4610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-cnv 4617  df-dm 4619  df-rn 4620
This theorem is referenced by:  rneqi  4837  rneqd  4838  xpima1  5055  feq1  5328  foeq1  5414  ixpsnf1o  6711
  Copyright terms: Public domain W3C validator