Theorem unidmrn 4963

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 4810	. . . 4 |- Rel `' A
2		relfld 4959	. . . 4 |- ( Rel `' A -> U. U. `' A = ( dom `' A u. ran `' A ) )
3	1, 2	ax-mp 7	. . 3 |- U. U. `' A = ( dom `' A u. ran `' A )
4	3	equncomi 3146	. 2 |- U. U. `' A = ( ran `' A u. dom `' A )
5		dfdm4 4628	. . 3 |- dom A = ran `' A
6		df-rn 4449	. . 3 |- ran A = dom `' A
7	5, 6	uneq12i 3152	. 2 |- ( dom A u. ran A ) = ( ran `' A u. dom `' A )
8	4, 7	eqtr4i 2111	1 |- U. U. `' A = ( dom A u. ran A )

Syntax hints:   = wceq 1289   u. cun 2997   U.cuni 3653   `'ccnv 4437   dom cdm 4438   ran crn 4439   Rel wrel 4443

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449

This theorem is referenced by:  relcnvfld  4964  dfdm2  4965