Theorem unidmrn 5234

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 5079	. . . 4 |- Rel `' A
2		relfld 5230	. . . 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 3327	. 2 |- U. U. `' A = ( ran `' A u. dom `' A )
5		dfdm4 4889	. . 3 |- dom A = ran `' A
6		df-rn 4704	. . 3 |- ran A = dom `' A
7	5, 6	uneq12i 3333	. 2 |- ( dom A u. ran A ) = ( ran `' A u. dom `' A )
8	4, 7	eqtr4i 2231	1 |- U. U. `' A = ( dom A u. ran A )

Syntax hints:   = wceq 1373   u. cun 3172   U.cuni 3864   `'ccnv 4692   dom cdm 4693   ran crn 4694   Rel wrel 4698

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  relcnvfld  5235  dfdm2  5236