Theorem unidmrn 6310

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 6134	. . . 4 ⊢ Rel ◡𝐴
2		relfld 6306	. . . 4 ⊢ (Rel ◡𝐴 → ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴))
3	1, 2	ax-mp 5	. . 3 ⊢ ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴)
4	3	equncomi 4183	. 2 ⊢ ∪ ∪ ◡𝐴 = (ran ◡𝐴 ∪ dom ◡𝐴)
5		dfdm4 5920	. . 3 ⊢ dom 𝐴 = ran ◡𝐴
6		df-rn 5711	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7	5, 6	uneq12i 4189	. 2 ⊢ (dom 𝐴 ∪ ran 𝐴) = (ran ◡𝐴 ∪ dom ◡𝐴)
8	4, 7	eqtr4i 2771	1 ⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴)

Syntax hints:   = wceq 1537   ∪ cun 3974  ∪ cuni 4931  ^◡ccnv 5699  dom cdm 5700  ran crn 5701  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711

This theorem is referenced by:  relcnvfld  6311  dfdm2  6312