Theorem cnvsng 5828

Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) (Proof shortened by BJ, 12-Feb-2022.)

Assertion

Ref	Expression
cnvsng	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Proof of Theorem cnvsng

Step	Hyp	Ref	Expression
1		cnvcnvsn 5824	. 2 ⊢ ◡◡{⟨𝐵, 𝐴⟩} = ◡{⟨𝐴, 𝐵⟩}
2		relsnopg 5426	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → Rel {⟨𝐵, 𝐴⟩})
3	2	ancoms 448	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {⟨𝐵, 𝐴⟩})
4		dfrel2 5794	. . 3 ⊢ (Rel {⟨𝐵, 𝐴⟩} ↔ ◡◡{⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
5	3, 4	sylib 209	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡◡{⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
6	1, 5	syl5eqr 2854	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1637   ∈ wcel 2156  {csn 4370  ⟨cop 4376  ^◡ccnv 5310  Rel wrel 5316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-xp 5317  df-rel 5318  df-cnv 5319

This theorem is referenced by:  cnvsn  5831  opswap  5836  funsng  6147  f1oprswap  6392