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Theorem clcllaw 42520
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 42515 . . . 4 clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
21bropaex12 5364 . . 3 ( clLaw 𝑀 → ( ∈ V ∧ 𝑀 ∈ V))
3 iscllaw 42518 . . . 4 (( ∈ V ∧ 𝑀 ∈ V) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
4 ovrspc2v 6872 . . . . 5 (((𝑋𝑀𝑌𝑀) ∧ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀) → (𝑋 𝑌) ∈ 𝑀)
54expcom 402 . . . 4 (∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀))
63, 5syl6bi 244 . . 3 (( ∈ V ∧ 𝑀 ∈ V) → ( clLaw 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)))
72, 6mpcom 38 . 2 ( clLaw 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀))
873impib 1144 1 (( clLaw 𝑀𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107  wcel 2155  wral 3055  Vcvv 3350   class class class wbr 4811  (class class class)co 6846   clLaw ccllaw 42512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-xp 5285  df-iota 6033  df-fv 6078  df-ov 6849  df-cllaw 42515
This theorem is referenced by: (None)
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