Theorem cnvsng 5089

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 3758	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	sneqd 3589	. . . 4 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
3	2	cnveqd 4780	. . 3 ⊢ (𝑥 = 𝐴 → ◡{⟨𝑥, 𝑦⟩} = ◡{⟨𝐴, 𝑦⟩})
4		opeq2 3759	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝐴⟩)
5	4	sneqd 3589	. . 3 ⊢ (𝑥 = 𝐴 → {⟨𝑦, 𝑥⟩} = {⟨𝑦, 𝐴⟩})
6	3, 5	eqeq12d 2180	. 2 ⊢ (𝑥 = 𝐴 → (◡{⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩} ↔ ◡{⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩}))
7		opeq2 3759	. . . . 5 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
8	7	sneqd 3589	. . . 4 ⊢ (𝑦 = 𝐵 → {⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
9	8	cnveqd 4780	. . 3 ⊢ (𝑦 = 𝐵 → ◡{⟨𝐴, 𝑦⟩} = ◡{⟨𝐴, 𝐵⟩})
10		opeq1 3758	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝑦, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
11	10	sneqd 3589	. . 3 ⊢ (𝑦 = 𝐵 → {⟨𝑦, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
12	9, 11	eqeq12d 2180	. 2 ⊢ (𝑦 = 𝐵 → (◡{⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩} ↔ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}))
13		vex 2729	. . 3 ⊢ 𝑥 ∈ V
14		vex 2729	. . 3 ⊢ 𝑦 ∈ V
15	13, 14	cnvsn 5086	. 2 ⊢ ◡{⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩}
16	6, 12, 15	vtocl2g 2790	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  {csn 3576  ⟨cop 3579  ^◡ccnv 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612

This theorem is referenced by:  opswapg  5090  funsng  5234