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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 5506 | . . 3 ⊢ ◡◡𝐵 = (𝐵 ↾ V) | |
2 | 1 | coeq2i 5204 | . 2 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ (𝐵 ↾ V)) |
3 | resco 5556 | . 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V)) | |
4 | relco 5550 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
5 | dfrel3 5510 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵)) | |
6 | 4, 5 | mpbi 219 | . 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵) |
7 | 2, 3, 6 | 3eqtr2i 2638 | 1 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 Vcvv 3173 ◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 Rel wrel 5043 |
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-ral 2901 df-rex 2902 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-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-res 5050 |
This theorem is referenced by: dfdm2 5584 cofunex2g 7024 trclubgNEW 36944 cnvtrrel 36981 trrelsuperrel2dg 36982 |
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