Theorem cnvimarndm 5100

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 5086	. 2 |- ( `' A " dom `' A ) = ran `' A
2		df-rn 4736	. . 3 |- ran A = dom `' A
3	2	imaeq2i 5074	. 2 |- ( `' A " ran A ) = ( `' A " dom `' A )
4		dfdm4 4923	. 2 |- dom A = ran `' A
5	1, 3, 4	3eqtr4i 2262	1 |- ( `' A " ran A ) = dom A

Syntax hints:   = wceq 1397   `'ccnv 4724   dom cdm 4725   ran crn 4726   "cima 4728

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  en2  6997  cnrest2  14959