Theorem rneq 4662

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 4610	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4638	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4449	. 2 |- ran A = dom `' A
4		df-rn 4449	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2145	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1289   `'ccnv 4437   dom cdm 4438   ran crn 4439

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-cnv 4446  df-dm 4448  df-rn 4449

This theorem is referenced by:  rneqi  4663  rneqd  4664  xpima1  4877  feq1  5145  foeq1  5229