Theorem rneq 4957

Description: Equality theorem for range. (Contributed by set.mm contributors, 29-Dec-1996.)

Assertion

Ref	Expression
rneq	|- (A = B -> ran A = ran B)

Proof of Theorem rneq

Step	Hyp	Ref	Expression
1		imaeq1 4938	. 2 |- (A = B -> (A"_V) = (B"_V))
2		df-rn 4787	. 2 |- ran A = (A"_V)
3		df-rn 4787	. 2 |- ran B = (B"_V)
4	1, 2, 3	3eqtr4g 2410	1 |- (A = B -> ran A = ran B)

Syntax hints:   -> wi 4   = wceq 1642  _Vcvv 2860  "cima 4723  ran crn 4774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rex 2621  df-br 4641  df-ima 4728  df-rn 4787

This theorem is referenced by:  rneqi  4958  rneqd  4959  feq1  5211  foeq1  5266  fconst5  5456  fvranfn  5870  map0e  6024  1cnc  6140  frecxpg  6316