Theorem cnvimarndm 4903

Description: The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)

Assertion

Ref	Expression
cnvimarndm	|- ( `' A " ran A ) = dom A

Proof of Theorem cnvimarndm

Step	Hyp	Ref	Expression
1		imadmrn 4891	. 2 |- ( `' A " dom `' A ) = ran `' A
2		df-rn 4550	. . 3 |- ran A = dom `' A
3	2	imaeq2i 4879	. 2 |- ( `' A " ran A ) = ( `' A " dom `' A )
4		dfdm4 4731	. 2 |- dom A = ran `' A
5	1, 3, 4	3eqtr4i 2170	1 |- ( `' A " ran A ) = dom A

Syntax hints:   = wceq 1331   `'ccnv 4538   dom cdm 4539   ran crn 4540   "cima 4542

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-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552

This theorem is referenced by:  cnrest2  12405