Theorem opswapg 5254

Description: Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)

Assertion

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

Proof of Theorem opswapg

Step	Hyp	Ref	Expression
1		cnvsng 5253	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
2	1	unieqd 3930	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ ◡{⟨𝐴, 𝐵⟩} = ∪ {⟨𝐵, 𝐴⟩})
3		elex 2827	. . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
4		elex 2827	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
5		opexg 4349	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ ∈ V)
6	3, 4, 5	syl2anr 290	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐵, 𝐴⟩ ∈ V)
7		unisng 3936	. . 3 ⊢ (⟨𝐵, 𝐴⟩ ∈ V → ∪ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩)
8	6, 7	syl 14	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩)
9	2, 8	eqtrd 2267	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815  {csn 3694  ⟨cop 3697  ∪ cuni 3919  ^◡ccnv 4753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762

This theorem is referenced by:  2nd1st  6387  cnvf1olem  6433  brtposg  6498  dftpos4  6507  tpostpos  6508  xpcomco  7090  fsumcnv  12148  fprodcnv  12336  txswaphmeolem  15311