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Theorem cores 5782
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 3354 . . . . . . 7 𝑧 ∈ V
2 vex 3354 . . . . . . 7 𝑦 ∈ V
31, 2brelrn 5494 . . . . . 6 (𝑧𝐵𝑦𝑦 ∈ ran 𝐵)
4 ssel 3746 . . . . . 6 (ran 𝐵𝐶 → (𝑦 ∈ ran 𝐵𝑦𝐶))
5 vex 3354 . . . . . . . 8 𝑥 ∈ V
65brres 5543 . . . . . . 7 (𝑦(𝐴𝐶)𝑥 ↔ (𝑦𝐴𝑥𝑦𝐶))
76rbaib 528 . . . . . 6 (𝑦𝐶 → (𝑦(𝐴𝐶)𝑥𝑦𝐴𝑥))
83, 4, 7syl56 36 . . . . 5 (ran 𝐵𝐶 → (𝑧𝐵𝑦 → (𝑦(𝐴𝐶)𝑥𝑦𝐴𝑥)))
98pm5.32d 566 . . . 4 (ran 𝐵𝐶 → ((𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥) ↔ (𝑧𝐵𝑦𝑦𝐴𝑥)))
109exbidv 2002 . . 3 (ran 𝐵𝐶 → (∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥) ↔ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)))
1110opabbidv 4850 . 2 (ran 𝐵𝐶 → {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥)} = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)})
12 df-co 5258 . 2 ((𝐴𝐶) ∘ 𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦(𝐴𝐶)𝑥)}
13 df-co 5258 . 2 (𝐴𝐵) = {⟨𝑧, 𝑥⟩ ∣ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥)}
1411, 12, 133eqtr4g 2830 1 (ran 𝐵𝐶 → ((𝐴𝐶) ∘ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wss 3723   class class class wbr 4786  {copab 4846  ran crn 5250  cres 5251  ccom 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261
This theorem is referenced by:  cocnvcnv1  5790  cores2  5792  relcoi2  5807  fco2  6199  fcoi2  6219  domss2  8275  canthp1lem2  9677  imasdsval2  16384  frmdss2  17608  gsumval3lem1  18513  gsumzres  18517  gsumzaddlem  18528  dprdf1  18640  kgencn2  21581  tsmsf1o  22168  lgamcvg2  25002  hhssims  28472  eulerpartgbij  30774  cvmlift2lem9a  31623  poimirlem9  33751  fourierdlem53  40893  funresfunco  41725
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