Theorem clintopcllaw 48199

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 48196	. . 3 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
2		ffnov 7475	. . . 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 6858	. . 3 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V)
6		iscllaw 48177	. . 3 ⊢ (( ⚬ ∈ ( clIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
7	5, 6	mpdan 687	. 2 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
8	4, 7	mpbird 257	1 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ clLaw 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   class class class wbr 5092   × cxp 5617   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   clLaw ccllaw 48171   clIntOp cclintop 48185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-map 8755  df-cllaw 48174  df-intop 48187  df-clintop 48188

This theorem is referenced by:  assintopcllaw  48200