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Theorem cores 5858
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 3389 . . . . . . 7 𝑧 ∈ V
2 vex 3389 . . . . . . 7 𝑦 ∈ V
31, 2brelrn 5561 . . . . . 6 (𝑧𝐵𝑦𝑦 ∈ ran 𝐵)
4 ssel 3793 . . . . . 6 (ran 𝐵𝐶 → (𝑦 ∈ ran 𝐵𝑦𝐶))
5 vex 3389 . . . . . . . 8 𝑥 ∈ V
65brresi 5610 . . . . . . 7 (𝑦(𝐴𝐶)𝑥 ↔ (𝑦𝐶𝑦𝐴𝑥))
76baib 532 . . . . . 6 (𝑦𝐶 → (𝑦(𝐴𝐶)𝑥𝑦𝐴𝑥))
83, 4, 7syl56 36 . . . . 5 (ran 𝐵𝐶 → (𝑧𝐵𝑦 → (𝑦(𝐴𝐶)𝑥𝑦𝐴𝑥)))
98pm5.32d 573 . . . 4 (ran 𝐵𝐶 → ((𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥) ↔ (𝑧𝐵𝑦𝑦𝐴𝑥)))
109exbidv 2017 . . 3 (ran 𝐵𝐶 → (∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)))
1110opabbidv 4910 . 2 (ran 𝐵𝐶 → {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥)} = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)})
12 df-co 5322 . 2 ((𝐴𝐶) ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥)}
13 df-co 5322 . 2 (𝐴𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)}
1411, 12, 133eqtr4g 2859 1 (ran 𝐵𝐶 → ((𝐴𝐶) ∘ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  wss 3770   class class class wbr 4844  {copab 4906  ran crn 5314  cres 5315  ccom 5317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-br 4845  df-opab 4907  df-xp 5319  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325
This theorem is referenced by:  cocnvcnv1  5866  cores2  5868  relcoi2  5883  funresfunco  6143  fco2  6275  fcoi2  6295  domss2  8362  canthp1lem2  9764  imasdsval2  16490  frmdss2  17715  gsumval3lem1  18620  gsumzres  18624  gsumzaddlem  18635  dprdf1  18747  kgencn2  21688  tsmsf1o  22275  lgamcvg2  25132  hhssims  28656  eulerpartgbij  30949  cvmlift2lem9a  31801  poimirlem9  33906  fourierdlem53  41114
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