Theorem relcnvfld 6271

Description: if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)

Assertion

Ref	Expression
relcnvfld	⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)

Proof of Theorem relcnvfld

Step	Hyp	Ref	Expression
1		relfld 6266	. 2 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2		unidmrn 6270	. 2 ⊢ ∪ ∪ ◡𝑅 = (dom 𝑅 ∪ ran 𝑅)
3	1, 2	eqtr4di 2818	1 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)

Syntax hints:   → wi 4   = wceq 1563   ∪ cun 3905  ∪ cuni 4868  ^◡ccnv 5651  dom cdm 5652  ran crn 5653  Rel wrel 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-rn 5663

This theorem is referenced by:  cnvps  18624  tsrdir  18650