Theorem unidmrn 5260

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

Assertion

Ref	Expression
unidmrn	⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴)

Proof of Theorem unidmrn

Step	Hyp	Ref	Expression
1		relcnv 5105	. . . 4 ⊢ Rel ◡𝐴
2		relfld 5256	. . . 4 ⊢ (Rel ◡𝐴 → ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴))
3	1, 2	ax-mp 5	. . 3 ⊢ ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴)
4	3	equncomi 3350	. 2 ⊢ ∪ ∪ ◡𝐴 = (ran ◡𝐴 ∪ dom ◡𝐴)
5		dfdm4 4914	. . 3 ⊢ dom 𝐴 = ran ◡𝐴
6		df-rn 4729	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7	5, 6	uneq12i 3356	. 2 ⊢ (dom 𝐴 ∪ ran 𝐴) = (ran ◡𝐴 ∪ dom ◡𝐴)
8	4, 7	eqtr4i 2253	1 ⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴)

Syntax hints:   = wceq 1395   ∪ cun 3195  ∪ cuni 3887  ^◡ccnv 4717  dom cdm 4718  ran crn 4719  Rel wrel 4723

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-dm 4728  df-rn 4729

This theorem is referenced by:  relcnvfld  5261  dfdm2  5262