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Mirrors > Home > MPE Home > Th. List > cnvun | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
cnvun | ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnv 5527 | . . 3 ⊢ ◡(𝐴 ∪ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} | |
2 | unopab 5109 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)} | |
3 | brun 5081 | . . . . 5 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)) | |
4 | 3 | opabbii 5097 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)} |
5 | 2, 4 | eqtr4i 2824 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} |
6 | 1, 5 | eqtr4i 2824 | . 2 ⊢ ◡(𝐴 ∪ 𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
7 | df-cnv 5527 | . . 3 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
8 | df-cnv 5527 | . . 3 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
9 | 7, 8 | uneq12i 4088 | . 2 ⊢ (◡𝐴 ∪ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
10 | 6, 9 | eqtr4i 2824 | 1 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 844 = wceq 1538 ∪ cun 3879 class class class wbr 5030 {copab 5092 ◡ccnv 5518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-br 5031 df-opab 5093 df-cnv 5527 |
This theorem is referenced by: rnun 5971 funcnvpr 6386 funcnvtp 6387 funcnvqp 6388 f1oun 6609 f1oprswap 6633 suppun 7833 sbthlem8 8618 domss2 8660 1sdom 8705 fsuppun 8836 fpwwe2lem13 10053 trclublem 14346 mbfres2 24249 ex-cnv 28222 cnvprop 30456 padct 30481 cycpmconjslem2 30847 eulerpartlemt 31739 mthmpps 32942 clcnvlem 40323 frege131d 40465 |
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