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Theorem cnvprop 32423
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 6215 . . . 4 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
21adantr 480 . . 3 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
3 cnvsng 6215 . . . 4 ((𝐶𝑉𝐷𝑊) → {⟨𝐶, 𝐷⟩} = {⟨𝐷, 𝐶⟩})
43adantl 481 . . 3 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → {⟨𝐶, 𝐷⟩} = {⟨𝐷, 𝐶⟩})
52, 4uneq12d 4159 . 2 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐷, 𝐶⟩}))
6 df-pr 4626 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
76cnveqi 5867 . . 3 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
8 cnvun 6135 . . 3 ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
97, 8eqtri 2754 . 2 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
10 df-pr 4626 . 2 {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩} = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐷, 𝐶⟩})
115, 9, 103eqtr4g 2791 1 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cun 3941  {csn 4623  {cpr 4625  cop 4629  ccnv 5668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677
This theorem is referenced by:  cycpm2tr  32782
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