Theorem unidmrn 5136

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 4982	. . . 4 |- Rel `' A
2		relfld 5132	. . . 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 3268	. 2 |- U. U. `' A = ( ran `' A u. dom `' A )
5		dfdm4 4796	. . 3 |- dom A = ran `' A
6		df-rn 4615	. . 3 |- ran A = dom `' A
7	5, 6	uneq12i 3274	. 2 |- ( dom A u. ran A ) = ( ran `' A u. dom `' A )
8	4, 7	eqtr4i 2189	1 |- U. U. `' A = ( dom A u. ran A )

Syntax hints:   = wceq 1343   u. cun 3114   U.cuni 3789   `'ccnv 4603   dom cdm 4604   ran crn 4605   Rel wrel 4609

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  relcnvfld  5137  dfdm2  5138