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Theorem cores2 5686
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 5348 . . . . . 6 dom 𝐴 = ran 𝐴
21sseq1i 3662 . . . . 5 (dom 𝐴𝐶 ↔ ran 𝐴𝐶)
3 cores 5676 . . . . 5 (ran 𝐴𝐶 → ((𝐵𝐶) ∘ 𝐴) = (𝐵𝐴))
42, 3sylbi 207 . . . 4 (dom 𝐴𝐶 → ((𝐵𝐶) ∘ 𝐴) = (𝐵𝐴))
5 cnvco 5340 . . . . 5 (𝐴(𝐵𝐶)) = ((𝐵𝐶) ∘ 𝐴)
6 cocnvcnv1 5684 . . . . 5 ((𝐵𝐶) ∘ 𝐴) = ((𝐵𝐶) ∘ 𝐴)
75, 6eqtri 2673 . . . 4 (𝐴(𝐵𝐶)) = ((𝐵𝐶) ∘ 𝐴)
8 cnvco 5340 . . . 4 (𝐴𝐵) = (𝐵𝐴)
94, 7, 83eqtr4g 2710 . . 3 (dom 𝐴𝐶(𝐴(𝐵𝐶)) = (𝐴𝐵))
109cnveqd 5330 . 2 (dom 𝐴𝐶(𝐴(𝐵𝐶)) = (𝐴𝐵))
11 relco 5671 . . 3 Rel (𝐴(𝐵𝐶))
12 dfrel2 5618 . . 3 (Rel (𝐴(𝐵𝐶)) ↔ (𝐴(𝐵𝐶)) = (𝐴(𝐵𝐶)))
1311, 12mpbi 220 . 2 (𝐴(𝐵𝐶)) = (𝐴(𝐵𝐶))
14 relco 5671 . . 3 Rel (𝐴𝐵)
15 dfrel2 5618 . . 3 (Rel (𝐴𝐵) ↔ (𝐴𝐵) = (𝐴𝐵))
1614, 15mpbi 220 . 2 (𝐴𝐵) = (𝐴𝐵)
1710, 13, 163eqtr3g 2708 1 (dom 𝐴𝐶 → (𝐴(𝐵𝐶)) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wss 3607  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  ccom 5147  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155
This theorem is referenced by:  fcoi1  6116  ofco2  20305
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