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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 1774 | . . . 4 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) | |
2 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | brco 5214 | . . . 4 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) |
5 | vex 3176 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
6 | 3, 5 | brcnv 5227 | . . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦) |
7 | 5, 2 | brcnv 5227 | . . . . . 6 ⊢ (𝑧◡𝐵𝑥 ↔ 𝑥𝐵𝑧) |
8 | 6, 7 | anbi12i 729 | . . . . 5 ⊢ ((𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ (𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) |
9 | 8 | exbii 1764 | . . . 4 ⊢ (∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧)) |
10 | 1, 4, 9 | 3bitr4i 291 | . . 3 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)) |
11 | 10 | opabbii 4649 | . 2 ⊢ {〈𝑦, 𝑥〉 ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} = {〈𝑦, 𝑥〉 ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)} |
12 | df-cnv 5046 | . 2 ⊢ ◡(𝐴 ∘ 𝐵) = {〈𝑦, 𝑥〉 ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} | |
13 | df-co 5047 | . 2 ⊢ (◡𝐵 ∘ ◡𝐴) = {〈𝑦, 𝑥〉 ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)} | |
14 | 11, 12, 13 | 3eqtr4i 2642 | 1 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∃wex 1695 class class class wbr 4583 {copab 4642 ◡ccnv 5037 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-co 5047 |
This theorem is referenced by: rncoss 5307 rncoeq 5310 dmco 5560 cores2 5565 co01 5567 coi2 5569 relcnvtr 5572 dfdm2 5584 f1co 6023 cofunex2g 7024 fparlem3 7166 fparlem4 7167 supp0cosupp0 7221 imacosupp 7222 fsuppcolem 8189 relexpcnv 13623 relexpaddg 13641 cnvps 17035 gimco 17533 gsumzf1o 18136 cnco 20880 ptrescn 21252 qtopcn 21327 hmeoco 21385 cncombf 23231 deg1val 23660 fcoinver 28798 ofpreima 28848 mbfmco 29653 eulerpartlemmf 29764 cvmliftmolem1 30517 cvmlift2lem9a 30539 cvmlift2lem9 30547 mclsppslem 30734 ftc1anclem3 32657 trlcocnv 35026 tendoicl 35102 cdlemk45 35253 cononrel1 36919 cononrel2 36920 cnvtrcl0 36952 cnvtrrel 36981 relexpaddss 37029 frege131d 37075 brco2f1o 37350 brco3f1o 37351 clsneicnv 37423 neicvgnvo 37433 smfco 39687 |
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