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Theorem cores2 5159
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 4837 . . . . . 6 dom 𝐴 = ran 𝐴
21sseq1i 3196 . . . . 5 (dom 𝐴𝐶 ↔ ran 𝐴𝐶)
3 cores 5150 . . . . 5 (ran 𝐴𝐶 → ((𝐵𝐶) ∘ 𝐴) = (𝐵𝐴))
42, 3sylbi 121 . . . 4 (dom 𝐴𝐶 → ((𝐵𝐶) ∘ 𝐴) = (𝐵𝐴))
5 cnvco 4830 . . . . 5 (𝐴(𝐵𝐶)) = ((𝐵𝐶) ∘ 𝐴)
6 cocnvcnv1 5157 . . . . 5 ((𝐵𝐶) ∘ 𝐴) = ((𝐵𝐶) ∘ 𝐴)
75, 6eqtri 2210 . . . 4 (𝐴(𝐵𝐶)) = ((𝐵𝐶) ∘ 𝐴)
8 cnvco 4830 . . . 4 (𝐴𝐵) = (𝐵𝐴)
94, 7, 83eqtr4g 2247 . . 3 (dom 𝐴𝐶(𝐴(𝐵𝐶)) = (𝐴𝐵))
109cnveqd 4821 . 2 (dom 𝐴𝐶(𝐴(𝐵𝐶)) = (𝐴𝐵))
11 relco 5145 . . 3 Rel (𝐴(𝐵𝐶))
12 dfrel2 5097 . . 3 (Rel (𝐴(𝐵𝐶)) ↔ (𝐴(𝐵𝐶)) = (𝐴(𝐵𝐶)))
1311, 12mpbi 145 . 2 (𝐴(𝐵𝐶)) = (𝐴(𝐵𝐶))
14 relco 5145 . . 3 Rel (𝐴𝐵)
15 dfrel2 5097 . . 3 (Rel (𝐴𝐵) ↔ (𝐴𝐵) = (𝐴𝐵))
1614, 15mpbi 145 . 2 (𝐴𝐵) = (𝐴𝐵)
1710, 13, 163eqtr3g 2245 1 (dom 𝐴𝐶 → (𝐴(𝐵𝐶)) = (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wss 3144  ccnv 4643  dom cdm 4644  ran crn 4645  cres 4646  ccom 4648  Rel wrel 4649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656
This theorem is referenced by:  cocnvres  5171  fcoi1  5415
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