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Theorem clcllaw 42152
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 42147 . . . 4 clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
21bropaex12 5226 . . 3 ( clLaw 𝑀 → ( ∈ V ∧ 𝑀 ∈ V))
3 iscllaw 42150 . . . 4 (( ∈ V ∧ 𝑀 ∈ V) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
4 ovrspc2v 6712 . . . . 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 1281 1 (( clLaw 𝑀𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wcel 2030  wral 2941  Vcvv 3231   class class class wbr 4685  (class class class)co 6690   clLaw ccllaw 42144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-iota 5889  df-fv 5934  df-ov 6693  df-cllaw 42147
This theorem is referenced by: (None)
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