Theorem releleccnv 34021

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 5503	. . 3 ⊢ Rel ◡𝑅
2		relelec 7787	. . 3 ⊢ (Rel ◡𝑅 → (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐵◡𝑅𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐵◡𝑅𝐴)
4		relbrcnvg 5504	. 2 ⊢ (Rel 𝑅 → (𝐵◡𝑅𝐴 ↔ 𝐴𝑅𝐵))
5	3, 4	syl5bb 272	1 ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]◡𝑅 ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 1990   class class class wbr 4653  ^◡ccnv 5113  Rel wrel 5119  [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by:  releccnveq  34022