![]() |
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 5832 | . . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V) | |
2 | 1 | coeq1i 5518 | . 2 ⊢ (◡◡𝐴 ∘ 𝐵) = ((𝐴 ↾ V) ∘ 𝐵) |
3 | ssv 3850 | . . 3 ⊢ ran 𝐵 ⊆ V | |
4 | cores 5883 | . . 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 2849 | 1 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 Vcvv 3414 ⊆ wss 3798 ◡ccnv 5345 ran crn 5347 ↾ cres 5348 ∘ ccom 5350 |
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 5007 ax-nul 5015 ax-pr 5129 |
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 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 |
This theorem is referenced by: cores2 5893 coires1 5898 cofunex2g 7398 mvdco 18222 deg1val 24262 trlcocnv 36790 trclubgNEW 38761 cnvtrrel 38798 trrelsuperrel2dg 38799 |
Copyright terms: Public domain | W3C validator |