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Mirrors > Home > ILE Home > Th. List > cnvco | GIF version |
Description: Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvco | ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exancom 1587 | . . . 4 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) | |
2 | vex 2689 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 2689 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | brco 4710 | . . . 4 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
5 | vex 2689 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
6 | 3, 5 | brcnv 4722 | . . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦) |
7 | 5, 2 | brcnv 4722 | . . . . . 6 ⊢ (𝑧◡𝐵𝑥 ↔ 𝑥𝐵𝑧) |
8 | 6, 7 | anbi12i 455 | . . . . 5 ⊢ ((𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ (𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) |
9 | 8 | exbii 1584 | . . . 4 ⊢ (∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) |
10 | 1, 4, 9 | 3bitr4i 211 | . . 3 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)) |
11 | 10 | opabbii 3995 | . 2 ⊢ {〈𝑦, 𝑥〉 ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} = {〈𝑦, 𝑥〉 ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)} |
12 | df-cnv 4547 | . 2 ⊢ ◡(𝐴 ∘ 𝐵) = {〈𝑦, 𝑥〉 ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} | |
13 | df-co 4548 | . 2 ⊢ (◡𝐵 ∘ ◡𝐴) = {〈𝑦, 𝑥〉 ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)} | |
14 | 11, 12, 13 | 3eqtr4i 2170 | 1 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 class class class wbr 3929 {copab 3988 ◡ccnv 4538 ∘ ccom 4543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-cnv 4547 df-co 4548 |
This theorem is referenced by: rncoss 4809 rncoeq 4812 dmco 5047 cores2 5051 co01 5053 coi2 5055 relcnvtr 5058 dfdm2 5073 f1co 5340 cofunex2g 6010 caseinj 6974 djuinj 6991 cnco 12390 hmeoco 12485 |
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