Theorem rneq 4813

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 4760	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4788	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4597	. 2 |- ran A = dom `' A
4		df-rn 4597	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2215	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1335   `'ccnv 4585   dom cdm 4586   ran crn 4587

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4594  df-dm 4596  df-rn 4597

This theorem is referenced by:  rneqi  4814  rneqd  4815  xpima1  5032  feq1  5302  foeq1  5388  ixpsnf1o  6681