Theorem rneq 4866

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

Step	Hyp	Ref	Expression
1		cnveq 4813	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4841	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4649	. 2 |- ran A = dom `' A
4		df-rn 4649	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2245	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1363   `'ccnv 4637   dom cdm 4638   ran crn 4639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-cnv 4646  df-dm 4648  df-rn 4649

This theorem is referenced by:  rneqi  4867  rneqd  4868  xpima1  5087  feq1  5360  foeq1  5446  ixpsnf1o  6749  imasex  12743