Theorem unidmrn 6271

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 6096	. . . 4 ⊢ Rel ◡𝐴
2		relfld 6267	. . . 4 ⊢ (Rel ◡𝐴 → ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴))
3	1, 2	ax-mp 5	. . 3 ⊢ ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴)
4	3	equncomi 4150	. 2 ⊢ ∪ ∪ ◡𝐴 = (ran ◡𝐴 ∪ dom ◡𝐴)
5		dfdm4 5888	. . 3 ⊢ dom 𝐴 = ran ◡𝐴
6		df-rn 5680	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7	5, 6	uneq12i 4156	. 2 ⊢ (dom 𝐴 ∪ ran 𝐴) = (ran ◡𝐴 ∪ dom ◡𝐴)
8	4, 7	eqtr4i 2757	1 ⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴)

Syntax hints:   = wceq 1533   ∪ cun 3941  ∪ cuni 4902  ^◡ccnv 5668  dom cdm 5669  ran crn 5670  Rel wrel 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680

This theorem is referenced by:  relcnvfld  6272  dfdm2  6273