Theorem unidmrn 5199

Description: The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)

Assertion

Ref	Expression
unidmrn	|- U. U. `' A = ( dom A u. ran A )

Proof of Theorem unidmrn

Step	Hyp	Ref	Expression
1		relcnv 5044	. . . 4 |- Rel `' A
2		relfld 5195	. . . 4 |- ( Rel `' A -> U. U. `' A = ( dom `' A u. ran `' A ) )
3	1, 2	ax-mp 5	. . 3 |- U. U. `' A = ( dom `' A u. ran `' A )
4	3	equncomi 3306	. 2 |- U. U. `' A = ( ran `' A u. dom `' A )
5		dfdm4 4855	. . 3 |- dom A = ran `' A
6		df-rn 4671	. . 3 |- ran A = dom `' A
7	5, 6	uneq12i 3312	. 2 |- ( dom A u. ran A ) = ( ran `' A u. dom `' A )
8	4, 7	eqtr4i 2217	1 |- U. U. `' A = ( dom A u. ran A )

Syntax hints:   = wceq 1364   u. cun 3152   U.cuni 3836   `'ccnv 4659   dom cdm 4660   ran crn 4661   Rel wrel 4665

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671

This theorem is referenced by:  relcnvfld  5200  dfdm2  5201