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Theorem clintopcllaw 47381
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 47378 . . 3 ( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀)
2 ffnov 7541 . . . 4 ( :(𝑀 × 𝑀)⟶𝑀 ↔ ( Fn (𝑀 × 𝑀) ∧ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
32simprbi 495 . . 3 ( :(𝑀 × 𝑀)⟶𝑀 → ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀)
41, 3syl 17 . 2 ( ∈ ( clIntOp ‘𝑀) → ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀)
5 elfvex 6928 . . 3 ( ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V)
6 iscllaw 47359 . . 3 (( ∈ ( clIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
75, 6mpdan 685 . 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 2098  wral 3051  Vcvv 3463   class class class wbr 5144   × cxp 5671   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7413   clLaw ccllaw 47353   clIntOp cclintop 47367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-map 8840  df-cllaw 47356  df-intop 47369  df-clintop 47370
This theorem is referenced by:  assintopcllaw  47382
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