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Mirrors > Home > ILE Home > Th. List > cocnvcnv2 | 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 4987 | . . 3 ⊢ ◡◡𝐵 = (𝐵 ↾ V) | |
2 | 1 | coeq2i 4694 | . 2 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ (𝐵 ↾ V)) |
3 | resco 5038 | . 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ (𝐵 ↾ V)) | |
4 | relco 5032 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
5 | dfrel3 4991 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵)) | |
6 | 4, 5 | mpbi 144 | . 2 ⊢ ((𝐴 ∘ 𝐵) ↾ V) = (𝐴 ∘ 𝐵) |
7 | 2, 3, 6 | 3eqtr2i 2164 | 1 ⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 Vcvv 2681 ◡ccnv 4533 ↾ cres 4536 ∘ ccom 4538 Rel wrel 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-res 4546 |
This theorem is referenced by: dfdm2 5068 cofunex2g 6003 |
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