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Theorem cores2 4776
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 4470 . . . . . 6 dom A = ran A
21sseq1i 2963 . . . . 5 (dom A𝐶 ↔ ran A𝐶)
3 cores 4767 . . . . 5 (ran A𝐶 → ((B𝐶) ∘ A) = (BA))
42, 3sylbi 114 . . . 4 (dom A𝐶 → ((B𝐶) ∘ A) = (BA))
5 cnvco 4463 . . . . 5 (A(B𝐶)) = ((B𝐶) ∘ A)
6 cocnvcnv1 4774 . . . . 5 ((B𝐶) ∘ A) = ((B𝐶) ∘ A)
75, 6eqtri 2057 . . . 4 (A(B𝐶)) = ((B𝐶) ∘ A)
8 cnvco 4463 . . . 4 (AB) = (BA)
94, 7, 83eqtr4g 2094 . . 3 (dom A𝐶(A(B𝐶)) = (AB))
109cnveqd 4454 . 2 (dom A𝐶(A(B𝐶)) = (AB))
11 relco 4762 . . 3 Rel (A(B𝐶))
12 dfrel2 4714 . . 3 (Rel (A(B𝐶)) ↔ (A(B𝐶)) = (A(B𝐶)))
1311, 12mpbi 133 . 2 (A(B𝐶)) = (A(B𝐶))
14 relco 4762 . . 3 Rel (AB)
15 dfrel2 4714 . . 3 (Rel (AB) ↔ (AB) = (AB))
1614, 15mpbi 133 . 2 (AB) = (AB)
1710, 13, 163eqtr3g 2092 1 (dom A𝐶 → (A(B𝐶)) = (AB))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wss 2911  ccnv 4287  dom cdm 4288  ran crn 4289  cres 4290  ccom 4292  Rel wrel 4293
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300
This theorem is referenced by:  fcoi1  5013
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