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Theorem cocnvcnv1 4831
Description: A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
Assertion
Ref Expression
cocnvcnv1 (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem cocnvcnv1
StepHypRef Expression
1 cnvcnv2 4774 . . 3 𝐴 = (𝐴 ↾ V)
21coeq1i 4495 . 2 (𝐴𝐵) = ((𝐴 ↾ V) ∘ 𝐵)
3 ssv 2965 . . 3 ran 𝐵 ⊆ V
4 cores 4824 . . 3 (ran 𝐵 ⊆ V → ((𝐴 ↾ V) ∘ 𝐵) = (𝐴𝐵))
53, 4ax-mp 7 . 2 ((𝐴 ↾ V) ∘ 𝐵) = (𝐴𝐵)
62, 5eqtri 2060 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1243  Vcvv 2557  wss 2917  ccnv 4344  ran crn 4346  cres 4347  ccom 4349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357
This theorem is referenced by:  cores2  4833  coires1  4838  cofunex2g  5739
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