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| Mirrors > Home > MPE Home > Th. List > cnvco | Structured version Visualization version 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 1861 | . . . 4 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) | |
| 2 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 3 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 4 | 2, 3 | brco 5881 | . . . 4 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
| 5 | vex 3484 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
| 6 | 3, 5 | brcnv 5893 | . . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦) |
| 7 | 5, 2 | brcnv 5893 | . . . . . 6 ⊢ (𝑧◡𝐵𝑥 ↔ 𝑥𝐵𝑧) |
| 8 | 6, 7 | anbi12i 628 | . . . . 5 ⊢ ((𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ (𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) |
| 9 | 8 | exbii 1848 | . . . 4 ⊢ (∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) |
| 10 | 1, 4, 9 | 3bitr4i 303 | . . 3 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)) |
| 11 | 10 | opabbii 5210 | . 2 ⊢ {〈𝑦, 𝑥〉 ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} = {〈𝑦, 𝑥〉 ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)} |
| 12 | df-cnv 5693 | . 2 ⊢ ◡(𝐴 ∘ 𝐵) = {〈𝑦, 𝑥〉 ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} | |
| 13 | df-co 5694 | . 2 ⊢ (◡𝐵 ∘ ◡𝐴) = {〈𝑦, 𝑥〉 ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)} | |
| 14 | 11, 12, 13 | 3eqtr4i 2775 | 1 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 class class class wbr 5143 {copab 5205 ◡ccnv 5684 ∘ ccom 5689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-co 5694 |
| This theorem is referenced by: rncoss 5986 rncoeq 5990 dmco 6274 cores2 6279 co01 6281 coi2 6283 relcnvtrg 6286 dfdm2 6301 f1cof1 6814 cofunex2g 7974 fparlem3 8139 fparlem4 8140 suppco 8231 fsuppcolem 9441 relexpcnv 15074 relexpaddg 15092 cnvps 18623 gimco 19286 gsumzf1o 19930 cnco 23274 ptrescn 23647 qtopcn 23722 hmeoco 23780 cncombf 25693 deg1val 26135 fcoinver 32617 ofpreima 32675 cycpmconjv 33162 cycpmconjs 33176 cyc3conja 33177 mbfmco 34266 eulerpartlemmf 34377 cvmliftmolem1 35286 cvmlift2lem9a 35308 cvmlift2lem9 35316 mclsppslem 35588 ftc1anclem3 37702 trlcocnv 40722 tendoicl 40798 cdlemk45 40949 rimco 42528 cononrel1 43607 cononrel2 43608 cnvtrcl0 43639 cnvtrrel 43683 relexpaddss 43731 frege131d 43777 brco2f1o 44045 brco3f1o 44046 clsneicnv 44118 neicvgnvo 44128 smfco 46817 |
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