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

Theorem cores2 4943
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 4628 . . . . . 6 dom 𝐴 = ran 𝐴
21sseq1i 3050 . . . . 5 (dom 𝐴𝐶 ↔ ran 𝐴𝐶)
3 cores 4934 . . . . 5 (ran 𝐴𝐶 → ((𝐵𝐶) ∘ 𝐴) = (𝐵𝐴))
42, 3sylbi 119 . . . 4 (dom 𝐴𝐶 → ((𝐵𝐶) ∘ 𝐴) = (𝐵𝐴))
5 cnvco 4621 . . . . 5 (𝐴(𝐵𝐶)) = ((𝐵𝐶) ∘ 𝐴)
6 cocnvcnv1 4941 . . . . 5 ((𝐵𝐶) ∘ 𝐴) = ((𝐵𝐶) ∘ 𝐴)
75, 6eqtri 2108 . . . 4 (𝐴(𝐵𝐶)) = ((𝐵𝐶) ∘ 𝐴)
8 cnvco 4621 . . . 4 (𝐴𝐵) = (𝐵𝐴)
94, 7, 83eqtr4g 2145 . . 3 (dom 𝐴𝐶(𝐴(𝐵𝐶)) = (𝐴𝐵))
109cnveqd 4612 . 2 (dom 𝐴𝐶(𝐴(𝐵𝐶)) = (𝐴𝐵))
11 relco 4929 . . 3 Rel (𝐴(𝐵𝐶))
12 dfrel2 4881 . . 3 (Rel (𝐴(𝐵𝐶)) ↔ (𝐴(𝐵𝐶)) = (𝐴(𝐵𝐶)))
1311, 12mpbi 143 . 2 (𝐴(𝐵𝐶)) = (𝐴(𝐵𝐶))
14 relco 4929 . . 3 Rel (𝐴𝐵)
15 dfrel2 4881 . . 3 (Rel (𝐴𝐵) ↔ (𝐴𝐵) = (𝐴𝐵))
1614, 15mpbi 143 . 2 (𝐴𝐵) = (𝐴𝐵)
1710, 13, 163eqtr3g 2143 1 (dom 𝐴𝐶 → (𝐴(𝐵𝐶)) = (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wss 2999  ccnv 4437  dom cdm 4438  ran crn 4439  cres 4440  ccom 4442  Rel wrel 4443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450
This theorem is referenced by:  cocnvres  4955  fcoi1  5191
  Copyright terms: Public domain W3C validator