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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 5828 | . . 3 ⊢ ◡◡𝐵 = (𝐵 ↾ V) | |
2 | 1 | coeq2i 5515 | . 2 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ (𝐵 ↾ V)) |
3 | resco 5880 | . 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V)) | |
4 | relco 5874 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
5 | dfrel3 5832 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵)) | |
6 | 4, 5 | mpbi 222 | . 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵) |
7 | 2, 3, 6 | 3eqtr2i 2855 | 1 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 Vcvv 3414 ◡ccnv 5341 ↾ cres 5344 ∘ ccom 5346 Rel wrel 5347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-res 5354 |
This theorem is referenced by: dfdm2 5908 cofunex2g 7393 trclubgNEW 38759 cnvtrrel 38796 trrelsuperrel2dg 38797 |
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