Theorem cnvprop 30432

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 6073	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
2	1	adantr 483	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
3		cnvsng 6073	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ◡{⟨𝐶, 𝐷⟩} = {⟨𝐷, 𝐶⟩})
4	3	adantl 484	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{⟨𝐶, 𝐷⟩} = {⟨𝐷, 𝐶⟩})
5	2, 4	uneq12d 4133	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩}) = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐷, 𝐶⟩}))
6		df-pr 4563	. . . 4 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
7	6	cnveqi 5738	. . 3 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
8		cnvun 5994	. . 3 ⊢ ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
9	7, 8	eqtri 2843	. 2 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
10		df-pr 4563	. 2 ⊢ {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩} = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐷, 𝐶⟩})
11	5, 9, 10	3eqtr4g 2880	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ∪ cun 3927  {csn 4560  {cpr 4562  ⟨cop 4566  ^◡ccnv 5547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-opab 5122  df-xp 5554  df-rel 5555  df-cnv 5556

This theorem is referenced by:  cycpm2tr  30782