Theorem clcllaw 44867

Description: Closure of a closed operation. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 21-Jan-2020.)

Assertion

Ref	Expression
clcllaw	⊢ (( ⚬ clLaw 𝑀 ∧ 𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)

Proof of Theorem clcllaw

Dummy variables 𝑚 𝑜 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cllaw 44862	. . . 4 ⊢ clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
2	1	bropaex12 5616	. . 3 ⊢ ( ⚬ clLaw 𝑀 → ( ⚬ ∈ V ∧ 𝑀 ∈ V))
3		iscllaw 44865	. . . 4 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ V) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
4		ovrspc2v 7182	. . . . 5 ⊢ (((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)
5	4	expcom 417	. . . 4 ⊢ (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀 → ((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀))
6	3, 5	syl6bi 256	. . 3 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ V) → ( ⚬ clLaw 𝑀 → ((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)))
7	2, 6	mpcom 38	. 2 ⊢ ( ⚬ clLaw 𝑀 → ((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀))
8	7	3impib 1113	1 ⊢ (( ⚬ clLaw 𝑀 ∧ 𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   class class class wbr 5036  (class class class)co 7156   clLaw ccllaw 44859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-xp 5534  df-iota 6299  df-fv 6348  df-ov 7159  df-cllaw 44862

This theorem is referenced by: (None)