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Theorem clcllaw 41614
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 41609 . . . 4 clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
21bropaex12 5100 . . 3 ( clLaw 𝑀 → ( ∈ V ∧ 𝑀 ∈ V))
3 iscllaw 41612 . . . 4 (( ∈ V ∧ 𝑀 ∈ V) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
4 ovrspc2v 6544 . . . . 5 (((𝑋𝑀𝑌𝑀) ∧ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀) → (𝑋 𝑌) ∈ 𝑀)
54expcom 449 . . . 4 (∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀))
63, 5syl6bi 241 . . 3 (( ∈ V ∧ 𝑀 ∈ V) → ( clLaw 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)))
72, 6mpcom 37 . 2 ( clLaw 𝑀 → ((𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀))
873impib 1253 1 (( clLaw 𝑀𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030  wcel 1975  wral 2890  Vcvv 3167   class class class wbr 4572  (class class class)co 6522   clLaw ccllaw 41606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-xp 5029  df-iota 5749  df-fv 5793  df-ov 6525  df-cllaw 41609
This theorem is referenced by: (None)
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