Theorem cnvsng 5151

Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)

Assertion

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

Proof of Theorem cnvsng

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3804	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	sneqd 3631	. . . 4 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
3	2	cnveqd 4838	. . 3 ⊢ (𝑥 = 𝐴 → ◡{⟨𝑥, 𝑦⟩} = ◡{⟨𝐴, 𝑦⟩})
4		opeq2 3805	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝐴⟩)
5	4	sneqd 3631	. . 3 ⊢ (𝑥 = 𝐴 → {⟨𝑦, 𝑥⟩} = {⟨𝑦, 𝐴⟩})
6	3, 5	eqeq12d 2208	. 2 ⊢ (𝑥 = 𝐴 → (◡{⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩} ↔ ◡{⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩}))
7		opeq2 3805	. . . . 5 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
8	7	sneqd 3631	. . . 4 ⊢ (𝑦 = 𝐵 → {⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
9	8	cnveqd 4838	. . 3 ⊢ (𝑦 = 𝐵 → ◡{⟨𝐴, 𝑦⟩} = ◡{⟨𝐴, 𝐵⟩})
10		opeq1 3804	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝑦, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
11	10	sneqd 3631	. . 3 ⊢ (𝑦 = 𝐵 → {⟨𝑦, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
12	9, 11	eqeq12d 2208	. 2 ⊢ (𝑦 = 𝐵 → (◡{⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩} ↔ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}))
13		vex 2763	. . 3 ⊢ 𝑥 ∈ V
14		vex 2763	. . 3 ⊢ 𝑦 ∈ V
15	13, 14	cnvsn 5148	. 2 ⊢ ◡{⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩}
16	6, 12, 15	vtocl2g 2824	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {csn 3618  ⟨cop 3621  ^◡ccnv 4658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667

This theorem is referenced by:  opswapg  5152  funsng  5300