Theorem rneq 4893

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 4840	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4868	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4674	. 2 |- ran A = dom `' A
4		df-rn 4674	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2254	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1364   `'ccnv 4662   dom cdm 4663   ran crn 4664

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by:  rneqi  4894  rneqd  4895  xpima1  5116  feq1  5390  foeq1  5476  ixpsnf1o  6795  imasex  12948