Theorem cnvun 5538

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 5122	. . 3 ⊢ ◡(𝐴 ∪ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥}
2		unopab 4728	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)}
3		brun 4703	. . . . 5 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
4	3	opabbii 4717	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)}
5	2, 4	eqtr4i 2647	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥}
6	1, 5	eqtr4i 2647	. 2 ⊢ ◡(𝐴 ∪ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
7		df-cnv 5122	. . 3 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
8		df-cnv 5122	. . 3 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
9	7, 8	uneq12i 3765	. 2 ⊢ (◡𝐴 ∪ ◡𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
10	6, 9	eqtr4i 2647	1 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)

Syntax hints:   ∨ wo 383   = wceq 1483   ∪ cun 3572   class class class wbr 4653  {copab 4712  ^◡ccnv 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-br 4654  df-opab 4713  df-cnv 5122

This theorem is referenced by:  rnun  5541  funcnvpr  5950  funcnvtp  5951  funcnvqp  5952  funcnvqpOLD  5953  f1oun  6156  f1oprswap  6180  suppun  7315  sbthlem8  8077  domss2  8119  1sdom  8163  fsuppun  8294  fpwwe2lem13  9464  trclublem  13734  strlemor1OLD  15969  xpsc  16217  mbfres2  23412  ex-cnv  27294  padct  29497  eulerpartlemt  30433  mthmpps  31479  clcnvlem  37930  frege131d  38056