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Theorem clintopcllaw 44939
Description: The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
Assertion
Ref Expression
clintopcllaw ( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)

Proof of Theorem clintopcllaw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clintop 44936 . . 3 ( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀)
2 ffnov 7293 . . . 4 ( :(𝑀 × 𝑀)⟶𝑀 ↔ ( Fn (𝑀 × 𝑀) ∧ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
32simprbi 500 . . 3 ( :(𝑀 × 𝑀)⟶𝑀 → ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀)
41, 3syl 17 . 2 ( ∈ ( clIntOp ‘𝑀) → ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀)
5 elfvex 6707 . . 3 ( ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V)
6 iscllaw 44917 . . 3 (( ∈ ( clIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
75, 6mpdan 687 . 2 ( ∈ ( clIntOp ‘𝑀) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
84, 7mpbird 260 1 ( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2114  wral 3053  Vcvv 3398   class class class wbr 5030   × cxp 5523   Fn wfn 6334  wf 6335  cfv 6339  (class class class)co 7170   clLaw ccllaw 44911   clIntOp cclintop 44925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-cllaw 44914  df-intop 44927  df-clintop 44928
This theorem is referenced by:  assintopcllaw  44940
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