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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvco1 | Structured version Visualization version GIF version |
Description: Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
Ref | Expression |
---|---|
cnvco1 | ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5745 | . 2 ⊢ Rel ◡(◡𝐴 ∘ 𝐵) | |
2 | relco 5875 | . 2 ⊢ Rel (◡𝐵 ∘ 𝐴) | |
3 | vex 3418 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
4 | vex 3418 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brcnv 5538 | . . . . . 6 ⊢ (𝑧◡𝐵𝑦 ↔ 𝑦𝐵𝑧) |
6 | 5 | bicomi 216 | . . . . 5 ⊢ (𝑦𝐵𝑧 ↔ 𝑧◡𝐵𝑦) |
7 | vex 3418 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 3, 7 | brcnv 5538 | . . . . 5 ⊢ (𝑧◡𝐴𝑥 ↔ 𝑥𝐴𝑧) |
9 | 6, 8 | anbi12ci 623 | . . . 4 ⊢ ((𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ (𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) |
10 | 9 | exbii 1949 | . . 3 ⊢ (∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) |
11 | 7, 4 | opelcnv 5537 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(◡𝐴 ∘ 𝐵) ↔ 〈𝑦, 𝑥〉 ∈ (◡𝐴 ∘ 𝐵)) |
12 | 4, 7 | opelco 5527 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ (◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥)) |
13 | 11, 12 | bitri 267 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥)) |
14 | 7, 4 | opelco 5527 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝐵 ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦)) |
15 | 10, 13, 14 | 3bitr4i 295 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡(◡𝐴 ∘ 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ (◡𝐵 ∘ 𝐴)) |
16 | 1, 2, 15 | eqrelriiv 5449 | 1 ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1658 ∃wex 1880 ∈ wcel 2166 〈cop 4404 class class class wbr 4874 ◡ccnv 5342 ∘ ccom 5347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 |
This theorem is referenced by: pprodcnveq 32530 |
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