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Theorem cnvun 4729
 Description: The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvun (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem cnvun
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnv 4353 . . 3 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴𝐵)𝑥}
2 unopab 3836 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥𝑦𝐵𝑥)}
3 brun 3810 . . . . 5 (𝑦(𝐴𝐵)𝑥 ↔ (𝑦𝐴𝑥𝑦𝐵𝑥))
43opabbii 3824 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴𝐵)𝑥} = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥𝑦𝐵𝑥)}
52, 4eqtr4i 2063 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴𝐵)𝑥}
61, 5eqtr4i 2063 . 2 (𝐴𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
7 df-cnv 4353 . . 3 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
8 df-cnv 4353 . . 3 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
97, 8uneq12i 3095 . 2 (𝐴𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
106, 9eqtr4i 2063 1 (𝐴𝐵) = (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∨ wo 629   = wceq 1243   ∪ cun 2915   class class class wbr 3764  {copab 3817  ◡ccnv 4344 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-br 3765  df-opab 3819  df-cnv 4353 This theorem is referenced by:  rnun  4732  f1oun  5146
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