Theorem cnvsng 5167

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 3818	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	sneqd 3645	. . . 4 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
3	2	cnveqd 4853	. . 3 ⊢ (𝑥 = 𝐴 → ◡{⟨𝑥, 𝑦⟩} = ◡{⟨𝐴, 𝑦⟩})
4		opeq2 3819	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝐴⟩)
5	4	sneqd 3645	. . 3 ⊢ (𝑥 = 𝐴 → {⟨𝑦, 𝑥⟩} = {⟨𝑦, 𝐴⟩})
6	3, 5	eqeq12d 2219	. 2 ⊢ (𝑥 = 𝐴 → (◡{⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩} ↔ ◡{⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩}))
7		opeq2 3819	. . . . 5 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
8	7	sneqd 3645	. . . 4 ⊢ (𝑦 = 𝐵 → {⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
9	8	cnveqd 4853	. . 3 ⊢ (𝑦 = 𝐵 → ◡{⟨𝐴, 𝑦⟩} = ◡{⟨𝐴, 𝐵⟩})
10		opeq1 3818	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝑦, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
11	10	sneqd 3645	. . 3 ⊢ (𝑦 = 𝐵 → {⟨𝑦, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
12	9, 11	eqeq12d 2219	. 2 ⊢ (𝑦 = 𝐵 → (◡{⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩} ↔ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}))
13		vex 2774	. . 3 ⊢ 𝑥 ∈ V
14		vex 2774	. . 3 ⊢ 𝑦 ∈ V
15	13, 14	cnvsn 5164	. 2 ⊢ ◡{⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩}
16	6, 12, 15	vtocl2g 2836	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372   ∈ wcel 2175  {csn 3632  ⟨cop 3635  ^◡ccnv 4673

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682

This theorem is referenced by:  opswapg  5168  funsng  5319