Theorem rneq 4924

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 4870	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4899	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4704	. 2 |- ran A = dom `' A
4		df-rn 4704	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2265	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1373   `'ccnv 4692   dom cdm 4693   ran crn 4694

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  rneqi  4925  rneqd  4926  xpima1  5148  feq1  5428  foeq1  5516  ixpsnf1o  6846  imasex  13252