MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cores Structured version   Visualization version   GIF version

Theorem cores 6255
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 3465 . . . . . . 7 𝑧 ∈ V
2 vex 3465 . . . . . . 7 𝑦 ∈ V
31, 2brelrn 5944 . . . . . 6 (𝑧𝐵𝑦𝑦 ∈ ran 𝐵)
4 ssel 3970 . . . . . 6 (ran 𝐵𝐶 → (𝑦 ∈ ran 𝐵𝑦𝐶))
5 vex 3465 . . . . . . . 8 𝑥 ∈ V
65brresi 5994 . . . . . . 7 (𝑦(𝐴𝐶)𝑥 ↔ (𝑦𝐶𝑦𝐴𝑥))
76baib 534 . . . . . 6 (𝑦𝐶 → (𝑦(𝐴𝐶)𝑥𝑦𝐴𝑥))
83, 4, 7syl56 36 . . . . 5 (ran 𝐵𝐶 → (𝑧𝐵𝑦 → (𝑦(𝐴𝐶)𝑥𝑦𝐴𝑥)))
98pm5.32d 575 . . . 4 (ran 𝐵𝐶 → ((𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥) ↔ (𝑧𝐵𝑦𝑦𝐴𝑥)))
109exbidv 1916 . . 3 (ran 𝐵𝐶 → (∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)))
1110opabbidv 5215 . 2 (ran 𝐵𝐶 → {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥)} = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)})
12 df-co 5687 . 2 ((𝐴𝐶) ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥)}
13 df-co 5687 . 2 (𝐴𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)}
1411, 12, 133eqtr4g 2790 1 (ran 𝐵𝐶 → ((𝐴𝐶) ∘ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  wss 3944   class class class wbr 5149  {copab 5211  ran crn 5679  cres 5680  ccom 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690
This theorem is referenced by:  cocnvcnv1  6263  cores2  6265  relcoi2  6283  funresfunco  6595  fco2  6750  fcoi2  6772  domss2  9161  cottrcl  9744  canthp1lem2  10678  imasdsval2  17501  frmdss2  18823  gsumval3lem1  19872  gsumzres  19876  gsumzaddlem  19888  dprdf1  20002  kgencn2  23505  tsmsf1o  24093  lgamcvg2  27032  hhssims  31156  ccatws1f1olast  32762  symgcom  32896  cycpmconjslem1  32967  cycpmconjslem2  32968  eulerpartgbij  34123  cvmlift2lem9a  35044  poimirlem9  37233  fourierdlem53  45685
  Copyright terms: Public domain W3C validator