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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 𝑀𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)

Proof of Theorem clcllaw
Dummy variables 𝑚 𝑜 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cllaw 44862 . . . 4 clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
21bropaex12 5616 . . 3 ( clLaw 𝑀 → ( ∈ V ∧ 𝑀 ∈ V))
3 iscllaw 44865 . . . 4 (( ∈ V ∧ 𝑀 ∈ V) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
4 ovrspc2v 7182 . . . . 5 (((𝑋𝑀𝑌𝑀) ∧ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀) → (𝑋 𝑌) ∈ 𝑀)
54expcom 417 . . . 4 (∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀))
63, 5syl6bi 256 . . 3 (( ∈ V ∧ 𝑀 ∈ V) → ( clLaw 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)))
72, 6mpcom 38 . 2 ( clLaw 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀))
873impib 1113 1 (( clLaw 𝑀𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)
 Colors of variables: wff setvar class 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)
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