Theorem rneq 4766

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 4713	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4741	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4550	. 2 |- ran A = dom `' A
4		df-rn 4550	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2197	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1331   `'ccnv 4538   dom cdm 4539   ran crn 4540

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550

This theorem is referenced by:  rneqi  4767  rneqd  4768  xpima1  4985  feq1  5255  foeq1  5341  ixpsnf1o  6630