Theorem cnvsng 6219

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 6215	. 2 ⊢ ◡◡{⟨𝐵, 𝐴⟩} = ◡{⟨𝐴, 𝐵⟩}
2		relsnopg 5784	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → Rel {⟨𝐵, 𝐴⟩})
3	2	ancoms 464	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {⟨𝐵, 𝐴⟩})
4		dfrel2 6182	. . 3 ⊢ (Rel {⟨𝐵, 𝐴⟩} ↔ ◡◡{⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
5	3, 4	sylib 221	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡◡{⟨𝐵, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
6	1, 5	eqtr3id 2809	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {csn 4584  ⟨cop 4590  ^◡ccnv 5654  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663

This theorem is used by:  cnvsn  6222  opswap  6225  funsng  6585  f1oprswap  6864  cnvprop  33171