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Theorem clcllaw 42355
 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 42350 . . . 4 clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
21bropaex12 5349 . . 3 ( clLaw 𝑀 → ( ∈ V ∧ 𝑀 ∈ V))
3 iscllaw 42353 . . . 4 (( ∈ V ∧ 𝑀 ∈ V) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
4 ovrspc2v 6836 . . . . 5 (((𝑋𝑀𝑌𝑀) ∧ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀) → (𝑋 𝑌) ∈ 𝑀)
54expcom 450 . . . 4 (∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀))
63, 5syl6bi 243 . . 3 (( ∈ V ∧ 𝑀 ∈ V) → ( clLaw 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)))
72, 6mpcom 38 . 2 ( clLaw 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀))
873impib 1109 1 (( clLaw 𝑀𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   ∈ wcel 2139  ∀wral 3050  Vcvv 3340   class class class wbr 4804  (class class class)co 6814   clLaw ccllaw 42347 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-iota 6012  df-fv 6057  df-ov 6817  df-cllaw 42350 This theorem is referenced by: (None)
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