Theorem unidmrn 6284

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 6108	. . . 4 ⊢ Rel ◡𝐴
2		relfld 6279	. . . 4 ⊢ (Rel ◡𝐴 → ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴))
3	1, 2	ax-mp 5	. . 3 ⊢ ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴)
4	3	equncomi 4114	. 2 ⊢ ∪ ∪ ◡𝐴 = (ran ◡𝐴 ∪ dom ◡𝐴)
5		dfdm4 5887	. . 3 ⊢ dom 𝐴 = ran ◡𝐴
6		df-rn 5674	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
7	5, 6	uneq12i 4120	. 2 ⊢ (dom 𝐴 ∪ ran 𝐴) = (ran ◡𝐴 ∪ dom ◡𝐴)
8	4, 7	eqtr4i 2791	1 ⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴)

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3904  ∪ cuni 4874  ^◡ccnv 5662  dom cdm 5663  ran crn 5664  Rel wrel 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674

This theorem is used by:  relcnvfld  6285  dfdm2  6286