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Mirrors > Home > NFE Home > Th. List > cnvun | GIF version |
Description: The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvun | ⊢ ◡(A ∪ B) = (◡A ∪ ◡B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unopab 4638 | . . 3 ⊢ ({〈x, y〉 ∣ yAx} ∪ {〈x, y〉 ∣ yBx}) = {〈x, y〉 ∣ (yAx ∨ yBx)} | |
2 | brun 4692 | . . . 4 ⊢ (y(A ∪ B)x ↔ (yAx ∨ yBx)) | |
3 | 2 | opabbii 4626 | . . 3 ⊢ {〈x, y〉 ∣ y(A ∪ B)x} = {〈x, y〉 ∣ (yAx ∨ yBx)} |
4 | 1, 3 | eqtr4i 2376 | . 2 ⊢ ({〈x, y〉 ∣ yAx} ∪ {〈x, y〉 ∣ yBx}) = {〈x, y〉 ∣ y(A ∪ B)x} |
5 | df-cnv 4785 | . . 3 ⊢ ◡A = {〈x, y〉 ∣ yAx} | |
6 | df-cnv 4785 | . . 3 ⊢ ◡B = {〈x, y〉 ∣ yBx} | |
7 | 5, 6 | uneq12i 3416 | . 2 ⊢ (◡A ∪ ◡B) = ({〈x, y〉 ∣ yAx} ∪ {〈x, y〉 ∣ yBx}) |
8 | df-cnv 4785 | . 2 ⊢ ◡(A ∪ B) = {〈x, y〉 ∣ y(A ∪ B)x} | |
9 | 4, 7, 8 | 3eqtr4ri 2384 | 1 ⊢ ◡(A ∪ B) = (◡A ∪ ◡B) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 357 = wceq 1642 ∪ cun 3207 {copab 4622 class class class wbr 4639 ◡ccnv 4771 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-opab 4623 df-br 4640 df-cnv 4785 |
This theorem is referenced by: rnun 5036 f1oun 5304 |
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