Theorem releleccnv 37968

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 8773	. . 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 282	1 ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2099   class class class wbr 5145  ^◡ccnv 5673  Rel wrel 5679  [cec 8724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-xp 5680  df-rel 5681  df-cnv 5682  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-ec 8728

This theorem is referenced by:  releccnveq  37969