Theorem cnvun 6131

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 5662	. . 3 ⊢ ◡(𝐴 ∪ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥}
2		unopab 5200	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)}
3		brun 5170	. . . . 5 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
4	3	opabbii 5186	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)}
5	2, 4	eqtr4i 2761	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥}
6	1, 5	eqtr4i 2761	. 2 ⊢ ◡(𝐴 ∪ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
7		df-cnv 5662	. . 3 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
8		df-cnv 5662	. . 3 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
9	7, 8	uneq12i 4141	. 2 ⊢ (◡𝐴 ∪ ◡𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
10	6, 9	eqtr4i 2761	1 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)

Syntax hints:   ∨ wo 847   = wceq 1540   ∪ cun 3924   class class class wbr 5119  {copab 5181  ^◡ccnv 5653

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-br 5120  df-opab 5182  df-cnv 5662

This theorem is referenced by:  rnun  6134  funcnvpr  6598  funcnvtp  6599  funcnvqp  6600  f1oun  6837  f1oprswap  6862  suppun  8183  sbthlem8  9104  domss2  9150  cnvfi  9190  1sdomOLD  9257  fsuppun  9399  fpwwe2lem12  10656  trclublem  15014  mbfres2  25598  ex-cnv  30418  suppun2  32661  cnvprop  32673  padct  32697  cycpmconjslem2  33166  eulerpartlemt  34403  mthmpps  35604  clcnvlem  43647  frege131d  43788