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Theorem cores 6205
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 3451 . . . . . . 7 𝑧 ∈ V
2 vex 3451 . . . . . . 7 𝑦 ∈ V
31, 2brelrn 5901 . . . . . 6 (𝑧𝐵𝑦𝑦 ∈ ran 𝐵)
4 ssel 3941 . . . . . 6 (ran 𝐵𝐶 → (𝑦 ∈ ran 𝐵𝑦𝐶))
5 vex 3451 . . . . . . . 8 𝑥 ∈ V
65brresi 5950 . . . . . . 7 (𝑦(𝐴𝐶)𝑥 ↔ (𝑦𝐶𝑦𝐴𝑥))
76baib 537 . . . . . 6 (𝑦𝐶 → (𝑦(𝐴𝐶)𝑥𝑦𝐴𝑥))
83, 4, 7syl56 36 . . . . 5 (ran 𝐵𝐶 → (𝑧𝐵𝑦 → (𝑦(𝐴𝐶)𝑥𝑦𝐴𝑥)))
98pm5.32d 578 . . . 4 (ran 𝐵𝐶 → ((𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥) ↔ (𝑧𝐵𝑦𝑦𝐴𝑥)))
109exbidv 1925 . . 3 (ran 𝐵𝐶 → (∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)))
1110opabbidv 5175 . 2 (ran 𝐵𝐶 → {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥)} = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)})
12 df-co 5646 . 2 ((𝐴𝐶) ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥)}
13 df-co 5646 . 2 (𝐴𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)}
1411, 12, 133eqtr4g 2798 1 (ran 𝐵𝐶 → ((𝐴𝐶) ∘ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wss 3914   class class class wbr 5109  {copab 5171  ran crn 5638  cres 5639  ccom 5641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649
This theorem is referenced by:  cocnvcnv1  6213  cores2  6215  relcoi2  6233  funresfunco  6546  fco2  6699  fcoi2  6721  domss2  9086  cottrcl  9663  canthp1lem2  10597  imasdsval2  17406  frmdss2  18681  gsumval3lem1  19690  gsumzres  19694  gsumzaddlem  19706  dprdf1  19820  kgencn2  22931  tsmsf1o  23519  lgamcvg2  26427  hhssims  30265  symgcom  31990  cycpmconjslem1  32059  cycpmconjslem2  32060  eulerpartgbij  33036  cvmlift2lem9a  33961  poimirlem9  36137  fourierdlem53  44490
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