Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  cores2 GIF version

Theorem cores2 4833
 Description: Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
Assertion
Ref Expression
cores2 (dom 𝐴𝐶 → (𝐴(𝐵𝐶)) = (𝐴𝐵))

Proof of Theorem cores2
StepHypRef Expression
1 dfdm4 4527 . . . . . 6 dom 𝐴 = ran 𝐴
21sseq1i 2969 . . . . 5 (dom 𝐴𝐶 ↔ ran 𝐴𝐶)
3 cores 4824 . . . . 5 (ran 𝐴𝐶 → ((𝐵𝐶) ∘ 𝐴) = (𝐵𝐴))
42, 3sylbi 114 . . . 4 (dom 𝐴𝐶 → ((𝐵𝐶) ∘ 𝐴) = (𝐵𝐴))
5 cnvco 4520 . . . . 5 (𝐴(𝐵𝐶)) = ((𝐵𝐶) ∘ 𝐴)
6 cocnvcnv1 4831 . . . . 5 ((𝐵𝐶) ∘ 𝐴) = ((𝐵𝐶) ∘ 𝐴)
75, 6eqtri 2060 . . . 4 (𝐴(𝐵𝐶)) = ((𝐵𝐶) ∘ 𝐴)
8 cnvco 4520 . . . 4 (𝐴𝐵) = (𝐵𝐴)
94, 7, 83eqtr4g 2097 . . 3 (dom 𝐴𝐶(𝐴(𝐵𝐶)) = (𝐴𝐵))
109cnveqd 4511 . 2 (dom 𝐴𝐶(𝐴(𝐵𝐶)) = (𝐴𝐵))
11 relco 4819 . . 3 Rel (𝐴(𝐵𝐶))
12 dfrel2 4771 . . 3 (Rel (𝐴(𝐵𝐶)) ↔ (𝐴(𝐵𝐶)) = (𝐴(𝐵𝐶)))
1311, 12mpbi 133 . 2 (𝐴(𝐵𝐶)) = (𝐴(𝐵𝐶))
14 relco 4819 . . 3 Rel (𝐴𝐵)
15 dfrel2 4771 . . 3 (Rel (𝐴𝐵) ↔ (𝐴𝐵) = (𝐴𝐵))
1614, 15mpbi 133 . 2 (𝐴𝐵) = (𝐴𝐵)
1710, 13, 163eqtr3g 2095 1 (dom 𝐴𝐶 → (𝐴(𝐵𝐶)) = (𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ⊆ wss 2917  ◡ccnv 4344  dom cdm 4345  ran crn 4346   ↾ cres 4347   ∘ ccom 4349  Rel wrel 4350 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:  fcoi1  5070
 Copyright terms: Public domain W3C validator