Theorem cnvsng 5187

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 3833	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	sneqd 3656	. . . 4 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
3	2	cnveqd 4872	. . 3 ⊢ (𝑥 = 𝐴 → ◡{⟨𝑥, 𝑦⟩} = ◡{⟨𝐴, 𝑦⟩})
4		opeq2 3834	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝐴⟩)
5	4	sneqd 3656	. . 3 ⊢ (𝑥 = 𝐴 → {⟨𝑦, 𝑥⟩} = {⟨𝑦, 𝐴⟩})
6	3, 5	eqeq12d 2222	. 2 ⊢ (𝑥 = 𝐴 → (◡{⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩} ↔ ◡{⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩}))
7		opeq2 3834	. . . . 5 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
8	7	sneqd 3656	. . . 4 ⊢ (𝑦 = 𝐵 → {⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
9	8	cnveqd 4872	. . 3 ⊢ (𝑦 = 𝐵 → ◡{⟨𝐴, 𝑦⟩} = ◡{⟨𝐴, 𝐵⟩})
10		opeq1 3833	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝑦, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
11	10	sneqd 3656	. . 3 ⊢ (𝑦 = 𝐵 → {⟨𝑦, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
12	9, 11	eqeq12d 2222	. 2 ⊢ (𝑦 = 𝐵 → (◡{⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩} ↔ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}))
13		vex 2779	. . 3 ⊢ 𝑥 ∈ V
14		vex 2779	. . 3 ⊢ 𝑦 ∈ V
15	13, 14	cnvsn 5184	. 2 ⊢ ◡{⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩}
16	6, 12, 15	vtocl2g 2842	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2178  {csn 3643  ⟨cop 3646  ^◡ccnv 4692

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701

This theorem is referenced by:  opswapg  5188  funsng  5339