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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 6045 | . . 3 ⊢ ◡◡𝐵 = (𝐵 ↾ V) | |
2 | 1 | coeq2i 5726 | . 2 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ (𝐵 ↾ V)) |
3 | resco 6098 | . 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V)) | |
4 | relco 6092 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
5 | dfrel3 6050 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵)) | |
6 | 4, 5 | mpbi 232 | . 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵) |
7 | 2, 3, 6 | 3eqtr2i 2850 | 1 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3495 ◡ccnv 5549 ↾ cres 5552 ∘ ccom 5554 Rel wrel 5555 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-res 5562 |
This theorem is referenced by: dfdm2 6127 cofunex2g 7645 trclubgNEW 39971 cnvtrrel 40008 trrelsuperrel2dg 40009 |
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