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Theorem caovcl 7627
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 1544 . 2
2 caovcl.1 . . . 4 ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
32adantl 481 . . 3 ((⊤ ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
43caovclg 7625 . 2 ((⊤ ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝑆)
51, 4mpan 690 1 ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wtru 1541  wcel 2108  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  ecopovtrn  8860  eceqoveq  8862  genpss  11044  genpnnp  11045  genpass  11049  expcllem  14113  txlly  23644  txnlly  23645
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