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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 1852 | . . . 4 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) | |
2 | vex 3495 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 3495 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | brco 5734 | . . . 4 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
5 | vex 3495 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
6 | 3, 5 | brcnv 5746 | . . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦) |
7 | 5, 2 | brcnv 5746 | . . . . . 6 ⊢ (𝑧◡𝐵𝑥 ↔ 𝑥𝐵𝑧) |
8 | 6, 7 | anbi12i 626 | . . . . 5 ⊢ ((𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ (𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) |
9 | 8 | exbii 1839 | . . . 4 ⊢ (∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) |
10 | 1, 4, 9 | 3bitr4i 304 | . . 3 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)) |
11 | 10 | opabbii 5124 | . 2 ⊢ {〈𝑦, 𝑥〉 ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} = {〈𝑦, 𝑥〉 ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)} |
12 | df-cnv 5556 | . 2 ⊢ ◡(𝐴 ∘ 𝐵) = {〈𝑦, 𝑥〉 ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} | |
13 | df-co 5557 | . 2 ⊢ (◡𝐵 ∘ ◡𝐴) = {〈𝑦, 𝑥〉 ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)} | |
14 | 11, 12, 13 | 3eqtr4i 2851 | 1 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∃wex 1771 class class class wbr 5057 {copab 5119 ◡ccnv 5547 ∘ ccom 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-cnv 5556 df-co 5557 |
This theorem is referenced by: rncoss 5836 rncoeq 5839 dmco 6100 cores2 6105 co01 6107 coi2 6109 relcnvtrg 6112 dfdm2 6125 f1co 6578 cofunex2g 7640 fparlem3 7798 fparlem4 7799 suppco 7859 supp0cosupp0OLD 7862 imacosuppOLD 7864 fsuppcolem 8852 relexpcnv 14382 relexpaddg 14400 cnvps 17810 gimco 18346 gsumzf1o 18961 cnco 21802 ptrescn 22175 qtopcn 22250 hmeoco 22308 cncombf 24186 deg1val 24617 fcoinver 30285 ofpreima 30338 cycpmconjv 30711 cycpmconjs 30725 cyc3conja 30726 mbfmco 31421 eulerpartlemmf 31532 cvmliftmolem1 32425 cvmlift2lem9a 32447 cvmlift2lem9 32455 mclsppslem 32727 ftc1anclem3 34850 trlcocnv 37736 tendoicl 37812 cdlemk45 37963 cononrel1 39832 cononrel2 39833 cnvtrcl0 39864 cnvtrrel 39893 relexpaddss 39941 frege131d 39987 brco2f1o 40260 brco3f1o 40261 clsneicnv 40333 neicvgnvo 40343 smfco 42954 |
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