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Theorem cores2 6079
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 5728 . . . . . 6 dom 𝐴 = ran 𝐴
21sseq1i 3943 . . . . 5 (dom 𝐴𝐶 ↔ ran 𝐴𝐶)
3 cores 6069 . . . . 5 (ran 𝐴𝐶 → ((𝐵𝐶) ∘ 𝐴) = (𝐵𝐴))
42, 3sylbi 220 . . . 4 (dom 𝐴𝐶 → ((𝐵𝐶) ∘ 𝐴) = (𝐵𝐴))
5 cnvco 5720 . . . . 5 (𝐴(𝐵𝐶)) = ((𝐵𝐶) ∘ 𝐴)
6 cocnvcnv1 6077 . . . . 5 ((𝐵𝐶) ∘ 𝐴) = ((𝐵𝐶) ∘ 𝐴)
75, 6eqtri 2821 . . . 4 (𝐴(𝐵𝐶)) = ((𝐵𝐶) ∘ 𝐴)
8 cnvco 5720 . . . 4 (𝐴𝐵) = (𝐵𝐴)
94, 7, 83eqtr4g 2858 . . 3 (dom 𝐴𝐶(𝐴(𝐵𝐶)) = (𝐴𝐵))
109cnveqd 5710 . 2 (dom 𝐴𝐶(𝐴(𝐵𝐶)) = (𝐴𝐵))
11 relco 6064 . . 3 Rel (𝐴(𝐵𝐶))
12 dfrel2 6013 . . 3 (Rel (𝐴(𝐵𝐶)) ↔ (𝐴(𝐵𝐶)) = (𝐴(𝐵𝐶)))
1311, 12mpbi 233 . 2 (𝐴(𝐵𝐶)) = (𝐴(𝐵𝐶))
14 relco 6064 . . 3 Rel (𝐴𝐵)
15 dfrel2 6013 . . 3 (Rel (𝐴𝐵) ↔ (𝐴𝐵) = (𝐴𝐵))
1614, 15mpbi 233 . 2 (𝐴𝐵) = (𝐴𝐵)
1710, 13, 163eqtr3g 2856 1 (dom 𝐴𝐶 → (𝐴(𝐵𝐶)) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wss 3881  ccnv 5518  dom cdm 5519  ran crn 5520  cres 5521  ccom 5523  Rel wrel 5524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531
This theorem is referenced by:  fcoi1  6526  ofco2  21056  cycpmconjvlem  30833
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