Theorem rneq 4838

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 4785	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4813	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4622	. 2 |- ran A = dom `' A
4		df-rn 4622	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2228	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1348   `'ccnv 4610   dom cdm 4611   ran crn 4612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by:  rneqi  4839  rneqd  4840  xpima1  5057  feq1  5330  foeq1  5416  ixpsnf1o  6714