Theorem rneq 4957

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 4902	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4931	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4734	. 2 |- ran A = dom `' A
4		df-rn 4734	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2287	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1395   `'ccnv 4722   dom cdm 4723   ran crn 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-cnv 4731  df-dm 4733  df-rn 4734

This theorem is referenced by:  rneqi  4958  rneqd  4959  xpima1  5181  feq1  5462  foeq1  5552  ixpsnf1o  6900  imasex  13378  ausgrusgrien  16010