Theorem cnvprop 33155

Description: Converse of a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)

Assertion

Ref	Expression
cnvprop	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩})

Proof of Theorem cnvprop

Step	Hyp	Ref	Expression
1		cnvsng 6223	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
2	1	adantr 486	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
3		cnvsng 6223	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ◡{⟨𝐶, 𝐷⟩} = {⟨𝐷, 𝐶⟩})
4	3	adantl 487	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{⟨𝐶, 𝐷⟩} = {⟨𝐷, 𝐶⟩})
5	2, 4	uneq12d 4119	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩}) = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐷, 𝐶⟩}))
6		df-pr 4590	. . . 4 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
7	6	cnveqi 5858	. . 3 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
8		cnvun 6137	. . 3 ⊢ ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
9	7, 8	eqtri 2785	. 2 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
10		df-pr 4590	. 2 ⊢ {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩} = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐷, 𝐶⟩})
11	5, 9, 10	3eqtr4g 2822	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∪ cun 3900  {csn 4587  {cpr 4589  ⟨cop 4593  ^◡ccnv 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667

This theorem is used by:  cycpm2tr  33546