Theorem cnvimarndm 5065

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 5051	. 2 |- ( `' A " dom `' A ) = ran `' A
2		df-rn 4704	. . 3 |- ran A = dom `' A
3	2	imaeq2i 5039	. 2 |- ( `' A " ran A ) = ( `' A " dom `' A )
4		dfdm4 4889	. 2 |- dom A = ran `' A
5	1, 3, 4	3eqtr4i 2238	1 |- ( `' A " ran A ) = dom A

Syntax hints:   = wceq 1373   `'ccnv 4692   dom cdm 4693   ran crn 4694   "cima 4696

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by:  en2  6936  cnrest2  14823