Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cocnvcnv1 | Structured version Visualization version GIF version |
Description: A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.) |
Ref | Expression |
---|---|
cocnvcnv1 | ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnv2 6052 | . . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V) | |
2 | 1 | coeq1i 5732 | . 2 ⊢ (◡◡𝐴 ∘ 𝐵) = ((𝐴 ↾ V) ∘ 𝐵) |
3 | ssv 3993 | . . 3 ⊢ ran 𝐵 ⊆ V | |
4 | cores 6104 | . . 3 ⊢ (ran 𝐵 ⊆ V → ((𝐴 ↾ V) ∘ 𝐵) = (𝐴 ∘ 𝐵)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ((𝐴 ↾ V) ∘ 𝐵) = (𝐴 ∘ 𝐵) |
6 | 2, 5 | eqtri 2846 | 1 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3496 ⊆ wss 3938 ◡ccnv 5556 ran crn 5558 ↾ cres 5559 ∘ ccom 5561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 |
This theorem is referenced by: cores2 6114 coires1 6119 cofunex2g 7653 mvdco 18575 deg1val 24692 trlcocnv 37858 trclubgNEW 39985 cnvtrrel 40022 trrelsuperrel2dg 40023 |
Copyright terms: Public domain | W3C validator |