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Theorem clintopcllaw 45405
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 45402 . . 3 ( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀)
2 ffnov 7401 . . . 4 ( :(𝑀 × 𝑀)⟶𝑀 ↔ ( Fn (𝑀 × 𝑀) ∧ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
32simprbi 497 . . 3 ( :(𝑀 × 𝑀)⟶𝑀 → ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀)
41, 3syl 17 . 2 ( ∈ ( clIntOp ‘𝑀) → ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀)
5 elfvex 6807 . . 3 ( ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V)
6 iscllaw 45383 . . 3 (( ∈ ( clIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
75, 6mpdan 684 . 2 ( ∈ ( clIntOp ‘𝑀) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
84, 7mpbird 256 1 ( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  wral 3064  Vcvv 3432   class class class wbr 5074   × cxp 5587   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275   clLaw ccllaw 45377   clIntOp cclintop 45391
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-cllaw 45380  df-intop 45393  df-clintop 45394
This theorem is referenced by:  assintopcllaw  45406
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