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Theorem cnvprop 32705
Description: Converse of a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
Assertion
Ref Expression
cnvprop (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩})

Proof of Theorem cnvprop
StepHypRef Expression
1 cnvsng 6243 . . . 4 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
21adantr 480 . . 3 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
3 cnvsng 6243 . . . 4 ((𝐶𝑉𝐷𝑊) → {⟨𝐶, 𝐷⟩} = {⟨𝐷, 𝐶⟩})
43adantl 481 . . 3 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → {⟨𝐶, 𝐷⟩} = {⟨𝐷, 𝐶⟩})
52, 4uneq12d 4169 . 2 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐷, 𝐶⟩}))
6 df-pr 4629 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
76cnveqi 5885 . . 3 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
8 cnvun 6162 . . 3 ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
97, 8eqtri 2765 . 2 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
10 df-pr 4629 . 2 {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩} = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐷, 𝐶⟩})
115, 9, 103eqtr4g 2802 1 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cun 3949  {csn 4626  {cpr 4628  cop 4632  ccnv 5684
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693
This theorem is referenced by:  cycpm2tr  33139
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