Theorem releleccnv 38929

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 6106	. . 3 ⊢ Rel ◡𝑅
2		relelec 8738	. . 3 ⊢ (Rel ◡𝑅 → (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐵◡𝑅𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐵◡𝑅𝐴)
4		relbrcnvg 6107	. 2 ⊢ (Rel 𝑅 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
5	3, 4	bitrid 286	1 ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2143   class class class wbr 5109  ^◡ccnv 5660  Rel wrel 5666  [cec 8688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692

This theorem is referenced by:  releccnveq  38930