Theorem rneq 4774

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 4721	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4749	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4558	. 2 |- ran A = dom `' A
4		df-rn 4558	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2198	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1332   `'ccnv 4546   dom cdm 4547   ran crn 4548

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by:  rneqi  4775  rneqd  4776  xpima1  4993  feq1  5263  foeq1  5349  ixpsnf1o  6638