Theorem relcnvfld 6172

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 6167	. 2 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2		unidmrn 6171	. 2 ⊢ ∪ ∪ ◡𝑅 = (dom 𝑅 ∪ ran 𝑅)
3	1, 2	eqtr4di 2797	1 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)

Syntax hints:   → wi 4   = wceq 1539   ∪ cun 3881  ∪ cuni 4836  ^◡ccnv 5579  dom cdm 5580  ran crn 5581  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  cnvps  18211  tsrdir  18237