Theorem releleccnv 38300

Description: Elementhood in a converse 𝑅-coset when 𝑅 is a relation. (Contributed by Peter Mazsa, 9-Dec-2018.)

Assertion

Ref	Expression
releleccnv	⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐴𝑅𝐵))

Proof of Theorem releleccnv

Step	Hyp	Ref	Expression
1		relcnv 6052	. . 3 ⊢ Rel ◡𝑅
2		relelec 8669	. . 3 ⊢ (Rel ◡𝑅 → (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐵◡𝑅𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐵◡𝑅𝐴)
4		relbrcnvg 6053	. 2 ⊢ (Rel 𝑅 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
5	3, 4	bitrid 283	1 ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2111   class class class wbr 5089  ^◡ccnv 5613  Rel wrel 5619  [cec 8620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ec 8624

This theorem is referenced by:  releccnveq  38301