Theorem rneq 4965

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 4910	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4939	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4742	. 2 |- ran A = dom `' A
4		df-rn 4742	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2289	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1398   `'ccnv 4730   dom cdm 4731   ran crn 4732

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by:  rneqi  4966  rneqd  4967  xpima1  5190  feq1  5472  foeq1  5564  ixpsnf1o  6948  imasex  13451  ausgrusgrien  16095  0uhgrsubgr  16189