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Theorem cores 6269
Description: Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cores (ran 𝐵𝐶 → ((𝐴𝐶) ∘ 𝐵) = (𝐴𝐵))

Proof of Theorem cores
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3484 . . . . . . 7 𝑧 ∈ V
2 vex 3484 . . . . . . 7 𝑦 ∈ V
31, 2brelrn 5953 . . . . . 6 (𝑧𝐵𝑦𝑦 ∈ ran 𝐵)
4 ssel 3977 . . . . . 6 (ran 𝐵𝐶 → (𝑦 ∈ ran 𝐵𝑦𝐶))
5 vex 3484 . . . . . . . 8 𝑥 ∈ V
65brresi 6006 . . . . . . 7 (𝑦(𝐴𝐶)𝑥 ↔ (𝑦𝐶𝑦𝐴𝑥))
76baib 535 . . . . . 6 (𝑦𝐶 → (𝑦(𝐴𝐶)𝑥𝑦𝐴𝑥))
83, 4, 7syl56 36 . . . . 5 (ran 𝐵𝐶 → (𝑧𝐵𝑦 → (𝑦(𝐴𝐶)𝑥𝑦𝐴𝑥)))
98pm5.32d 577 . . . 4 (ran 𝐵𝐶 → ((𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥) ↔ (𝑧𝐵𝑦𝑦𝐴𝑥)))
109exbidv 1921 . . 3 (ran 𝐵𝐶 → (∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)))
1110opabbidv 5209 . 2 (ran 𝐵𝐶 → {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥)} = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)})
12 df-co 5694 . 2 ((𝐴𝐶) ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥)}
13 df-co 5694 . 2 (𝐴𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)}
1411, 12, 133eqtr4g 2802 1 (ran 𝐵𝐶 → ((𝐴𝐶) ∘ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wss 3951   class class class wbr 5143  {copab 5205  ran crn 5686  cres 5687  ccom 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697
This theorem is referenced by:  cocnvcnv1  6277  cores2  6279  relcoi2  6297  funresfunco  6607  fco2  6762  fcoi2  6783  f1ocoima  7323  domss2  9176  cottrcl  9759  canthp1lem2  10693  imasdsval2  17561  frmdss2  18876  gsumval3lem1  19923  gsumzres  19927  gsumzaddlem  19939  dprdf1  20053  kgencn2  23565  tsmsf1o  24153  lgamcvg2  27098  hhssims  31293  ccatws1f1olast  32937  symgcom  33103  cycpmconjslem1  33174  cycpmconjslem2  33175  eulerpartgbij  34374  cvmlift2lem9a  35308  poimirlem9  37636  fourierdlem53  46174  tposres3  48781
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