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Theorem rneq 4650
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 4598 . . 3 (𝐴 = 𝐵𝐴 = 𝐵)
21dmeqd 4626 . 2 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3 df-rn 4439 . 2 ran 𝐴 = dom 𝐴
4 df-rn 4439 . 2 ran 𝐵 = dom 𝐵
52, 3, 43eqtr4g 2145 1 (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  ccnv 4427  dom cdm 4428  ran crn 4429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by:  rneqi  4651  rneqd  4652  xpima1  4864  feq1  5131  foeq1  5213
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