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Theorem caovcl 7596
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 1546 . 2
2 caovcl.1 . . . 4 ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
32adantl 483 . . 3 ((⊤ ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
43caovclg 7594 . 2 ((⊤ ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝑆)
51, 4mpan 689 1 ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wtru 1543  wcel 2107  (class class class)co 7404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7407
This theorem is referenced by:  ecopovtrn  8810  eceqoveq  8812  genpss  10995  genpnnp  10996  genpass  11000  expcllem  14034  txlly  23122  txnlly  23123
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