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Theorem caovcl 5815
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 1294 . 2
2 caovcl.1 . . . 4 ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
32adantl 272 . . 3 ((⊤ ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
43caovclg 5813 . 2 ((⊤ ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝑆)
51, 4mpan 416 1 ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wtru 1291  wcel 1439  (class class class)co 5668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-iota 4995  df-fv 5038  df-ov 5671
This theorem is referenced by:  ecopovtrn  6405  ecopovtrng  6408  genpelvl  7134  genpelvu  7135  genpml  7139  genpmu  7140  genprndl  7143  genprndu  7144  genpassl  7146  genpassu  7147  genpassg  7148  expcllem  10029
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