Theorem clintopcllaw 45293

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.

Step	Hyp	Ref	Expression
1		clintop 45290	. . 3 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
2		ffnov 7379	. . . 4 ⊢ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ↔ ( ⚬ Fn (𝑀 × 𝑀) ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
3	2	simprbi 496	. . 3 ⊢ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀)
4	1, 3	syl 17	. 2 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀)
5		elfvex 6789	. . 3 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V)
6		iscllaw 45271	. . 3 ⊢ (( ⚬ ∈ ( clIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
7	5, 6	mpdan 683	. 2 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
8	4, 7	mpbird 256	1 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ clLaw 𝑀)

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   class class class wbr 5070   × cxp 5578   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   clLaw ccllaw 45265   clIntOp cclintop 45279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-cllaw 45268  df-intop 45281  df-clintop 45282

This theorem is referenced by:  assintopcllaw  45294