Theorem cnvsng 5106

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 3774	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	sneqd 3602	. . . 4 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
3	2	cnveqd 4796	. . 3 ⊢ (𝑥 = 𝐴 → ◡{⟨𝑥, 𝑦⟩} = ◡{⟨𝐴, 𝑦⟩})
4		opeq2 3775	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝐴⟩)
5	4	sneqd 3602	. . 3 ⊢ (𝑥 = 𝐴 → {⟨𝑦, 𝑥⟩} = {⟨𝑦, 𝐴⟩})
6	3, 5	eqeq12d 2190	. 2 ⊢ (𝑥 = 𝐴 → (◡{⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩} ↔ ◡{⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩}))
7		opeq2 3775	. . . . 5 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
8	7	sneqd 3602	. . . 4 ⊢ (𝑦 = 𝐵 → {⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
9	8	cnveqd 4796	. . 3 ⊢ (𝑦 = 𝐵 → ◡{⟨𝐴, 𝑦⟩} = ◡{⟨𝐴, 𝐵⟩})
10		opeq1 3774	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝑦, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
11	10	sneqd 3602	. . 3 ⊢ (𝑦 = 𝐵 → {⟨𝑦, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
12	9, 11	eqeq12d 2190	. 2 ⊢ (𝑦 = 𝐵 → (◡{⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩} ↔ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}))
13		vex 2738	. . 3 ⊢ 𝑥 ∈ V
14		vex 2738	. . 3 ⊢ 𝑦 ∈ V
15	13, 14	cnvsn 5103	. 2 ⊢ ◡{⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩}
16	6, 12, 15	vtocl2g 2799	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2146  {csn 3589  ⟨cop 3592  ^◡ccnv 4619

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628

This theorem is referenced by:  opswapg  5107  funsng  5254