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Theorem rneq 4504
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 4452 . . 3 (A = BA = B)
21dmeqd 4480 . 2 (A = B → dom A = dom B)
3 df-rn 4299 . 2 ran A = dom A
4 df-rn 4299 . 2 ran B = dom B
52, 3, 43eqtr4g 2094 1 (A = B → ran A = ran B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  ccnv 4287  dom cdm 4288  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-bndl 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:  rneqi  4505  rneqd  4506  xpima1  4710  feq1  4973  foeq1  5045
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