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Mirrors > Home > MPE Home > Th. List > cocnvcnv2 | Structured version Visualization version GIF version |
Description: A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.) |
Ref | Expression |
---|---|
cocnvcnv2 | ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnv2 6215 | . . 3 ⊢ ◡◡𝐵 = (𝐵 ↾ V) | |
2 | 1 | coeq2i 5874 | . 2 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ (𝐵 ↾ V)) |
3 | resco 6272 | . 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V)) | |
4 | relco 6129 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
5 | dfrel3 6220 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵)) | |
6 | 4, 5 | mpbi 230 | . 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵) |
7 | 2, 3, 6 | 3eqtr2i 2769 | 1 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3478 ◡ccnv 5688 ↾ cres 5691 ∘ ccom 5693 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-res 5701 |
This theorem is referenced by: dfdm2 6303 cofunex2g 7973 trclubgNEW 43608 cnvtrrel 43660 trrelsuperrel2dg 43661 |
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