Theorem cnvsng 6189

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 6185	. 2 ⊢ ◡◡{⟨𝐵, 𝐴⟩} = ◡{⟨𝐴, 𝐵⟩}
2		relsnopg 5760	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → Rel {⟨𝐵, 𝐴⟩})
3	2	ancoms 458	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {⟨𝐵, 𝐴⟩})
4		dfrel2 6155	. . 3 ⊢ (Rel {⟨𝐵, 𝐴⟩} ↔ ◡◡{⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
5	3, 4	sylib 218	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡◡{⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
6	1, 5	eqtr3id 2786	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4582  ⟨cop 4588  ^◡ccnv 5631  Rel wrel 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640

This theorem is referenced by:  cnvsn  6192  opswap  6195  funsng  6551  f1oprswap  6827  cnvprop  32786