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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 1572 | . . . 4 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) | |
2 | vex 2663 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 2663 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | brco 4680 | . . . 4 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
5 | vex 2663 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
6 | 3, 5 | brcnv 4692 | . . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦) |
7 | 5, 2 | brcnv 4692 | . . . . . 6 ⊢ (𝑧◡𝐵𝑥 ↔ 𝑥𝐵𝑧) |
8 | 6, 7 | anbi12i 455 | . . . . 5 ⊢ ((𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ (𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) |
9 | 8 | exbii 1569 | . . . 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 3965 | . 2 ⊢ {〈𝑦, 𝑥〉 ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} = {〈𝑦, 𝑥〉 ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)} |
12 | df-cnv 4517 | . 2 ⊢ ◡(𝐴 ∘ 𝐵) = {〈𝑦, 𝑥〉 ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} | |
13 | df-co 4518 | . 2 ⊢ (◡𝐵 ∘ ◡𝐴) = {〈𝑦, 𝑥〉 ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)} | |
14 | 11, 12, 13 | 3eqtr4i 2148 | 1 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∃wex 1453 class class class wbr 3899 {copab 3958 ◡ccnv 4508 ∘ ccom 4513 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-cnv 4517 df-co 4518 |
This theorem is referenced by: rncoss 4779 rncoeq 4782 dmco 5017 cores2 5021 co01 5023 coi2 5025 relcnvtr 5028 dfdm2 5043 f1co 5310 cofunex2g 5978 caseinj 6942 djuinj 6959 cnco 12317 hmeoco 12412 |
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