Theorem unidmrn 6230

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 6056	. . . 4 ⊢ Rel ◡𝐴
2		relfld 6226	. . . 4 ⊢ (Rel ◡𝐴 → ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴))
3	1, 2	ax-mp 5	. . 3 ⊢ ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴)
4	3	equncomi 4090	. 2 ⊢ ∪ ∪ ◡𝐴 = (ran ◡𝐴 ∪ dom ◡𝐴)
5		dfdm4 5837	. . 3 ⊢ dom 𝐴 = ran ◡𝐴
6		df-rn 5629	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7	5, 6	uneq12i 4096	. 2 ⊢ (dom 𝐴 ∪ ran 𝐴) = (ran ◡𝐴 ∪ dom ◡𝐴)
8	4, 7	eqtr4i 2765	1 ⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴)

Syntax hints:   = wceq 1547   ∪ cun 3881  ∪ cuni 4838  ^◡ccnv 5617  dom cdm 5618  ran crn 5619  Rel wrel 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628  df-rn 5629

This theorem is referenced by:  relcnvfld  6231  dfdm2  6232