ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caovcl GIF version

Theorem caovcl 5708
Description: Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
Hypothesis
Ref Expression
caovcl.1 ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
Assertion
Ref Expression
caovcl ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem caovcl
StepHypRef Expression
1 tru 1289 . 2
2 caovcl.1 . . . 4 ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
32adantl 271 . . 3 ((⊤ ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
43caovclg 5706 . 2 ((⊤ ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝑆)
51, 4mpan 415 1 ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wtru 1286  wcel 1434  (class class class)co 5565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-un 2987  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-iota 4918  df-fv 4961  df-ov 5568
This theorem is referenced by:  ecopovtrn  6292  ecopovtrng  6295  genpelvl  6841  genpelvu  6842  genpml  6846  genpmu  6847  genprndl  6850  genprndu  6851  genpassl  6853  genpassu  6854  genpassg  6855  expcllem  9661
  Copyright terms: Public domain W3C validator