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Theorem clintopcllaw 46231
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 46228 . . 3 ( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀)
2 ffnov 7484 . . . 4 ( :(𝑀 × 𝑀)⟶𝑀 ↔ ( Fn (𝑀 × 𝑀) ∧ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
32simprbi 498 . . 3 ( :(𝑀 × 𝑀)⟶𝑀 → ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀)
41, 3syl 17 . 2 ( ∈ ( clIntOp ‘𝑀) → ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀)
5 elfvex 6881 . . 3 ( ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V)
6 iscllaw 46209 . . 3 (( ∈ ( clIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
75, 6mpdan 686 . 2 ( ∈ ( clIntOp ‘𝑀) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
84, 7mpbird 257 1 ( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107  wral 3061  Vcvv 3444   class class class wbr 5106   × cxp 5632   Fn wfn 6492  wf 6493  cfv 6497  (class class class)co 7358   clLaw ccllaw 46203   clIntOp cclintop 46217
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-cllaw 46206  df-intop 46219  df-clintop 46220
This theorem is referenced by:  assintopcllaw  46232
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