Theorem unidmrn 5079

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 4925	. . . 4 |- Rel `' A
2		relfld 5075	. . . 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 3227	. 2 |- U. U. `' A = ( ran `' A u. dom `' A )
5		dfdm4 4739	. . 3 |- dom A = ran `' A
6		df-rn 4558	. . 3 |- ran A = dom `' A
7	5, 6	uneq12i 3233	. 2 |- ( dom A u. ran A ) = ( ran `' A u. dom `' A )
8	4, 7	eqtr4i 2164	1 |- U. U. `' A = ( dom A u. ran A )

Syntax hints:   = wceq 1332   u. cun 3074   U.cuni 3744   `'ccnv 4546   dom cdm 4547   ran crn 4548   Rel wrel 4552

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by:  relcnvfld  5080  dfdm2  5081