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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 5563 | . . 3 ⊢ ◡(𝐴 ∪ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} | |
2 | unopab 5145 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)} | |
3 | brun 5117 | . . . . 5 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)) | |
4 | 3 | opabbii 5133 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} = {〈𝑥, 𝑦〉 ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)} |
5 | 2, 4 | eqtr4i 2847 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) = {〈𝑥, 𝑦〉 ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} |
6 | 1, 5 | eqtr4i 2847 | . 2 ⊢ ◡(𝐴 ∪ 𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
7 | df-cnv 5563 | . . 3 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
8 | df-cnv 5563 | . . 3 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
9 | 7, 8 | uneq12i 4137 | . 2 ⊢ (◡𝐴 ∪ ◡𝐵) = ({〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ∪ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
10 | 6, 9 | eqtr4i 2847 | 1 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1537 ∪ cun 3934 class class class wbr 5066 {copab 5128 ◡ccnv 5554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3941 df-br 5067 df-opab 5129 df-cnv 5563 |
This theorem is referenced by: rnun 6004 funcnvpr 6416 funcnvtp 6417 funcnvqp 6418 f1oun 6634 f1oprswap 6658 suppun 7850 sbthlem8 8634 domss2 8676 1sdom 8721 fsuppun 8852 fpwwe2lem13 10064 trclublem 14355 mbfres2 24246 ex-cnv 28216 cnvprop 30432 padct 30455 cycpmconjslem2 30797 eulerpartlemt 31629 mthmpps 32829 clcnvlem 40003 frege131d 40129 |
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