Theorem cnvun 5016

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	⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)

Proof of Theorem cnvun

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnv 4619	. . 3 ⊢ ◡(𝐴 ∪ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥}
2		unopab 4068	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)}
3		brun 4040	. . . . 5 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
4	3	opabbii 4056	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)}
5	2, 4	eqtr4i 2194	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥}
6	1, 5	eqtr4i 2194	. 2 ⊢ ◡(𝐴 ∪ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
7		df-cnv 4619	. . 3 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
8		df-cnv 4619	. . 3 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
9	7, 8	uneq12i 3279	. 2 ⊢ (◡𝐴 ∪ ◡𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
10	6, 9	eqtr4i 2194	1 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)

Syntax hints:   ∨ wo 703   = wceq 1348   ∪ cun 3119   class class class wbr 3989  {copab 4049  ^◡ccnv 4610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-br 3990  df-opab 4051  df-cnv 4619

This theorem is referenced by:  rnun  5019  f1oun  5462  sbthlemi8  6941  caseinj  7066  djuinj  7083