Theorem clcllaw 48682

Description: Closure of a closed operation. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 21-Jan-2020.)

Assertion

Ref	Expression
clcllaw	⊢ (( ⚬ clLaw 𝑀 ∧ 𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)

Proof of Theorem clcllaw

Dummy variables 𝑚 𝑜 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cllaw 48677	. . . 4 ⊢ clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
2	1	bropaex12 5709	. . 3 ⊢ ( ⚬ clLaw 𝑀 → ( ⚬ ∈ V ∧ 𝑀 ∈ V))
3		iscllaw 48680	. . . 4 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ V) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
4		ovrspc2v 7382	. . . . 5 ⊢ (((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)
5	4	expcom 414	. . . 4 ⊢ (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀 → ((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀))
6	3, 5	biimtrdi 254	. . 3 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ V) → ( ⚬ clLaw 𝑀 → ((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)))
7	2, 6	mpcom 38	. 2 ⊢ ( ⚬ clLaw 𝑀 → ((𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀))
8	7	3impib 1122	1 ⊢ (( ⚬ clLaw 𝑀 ∧ 𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   class class class wbr 5072  (class class class)co 7356   clLaw ccllaw 48674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-iota 6441  df-fv 6493  df-ov 7359  df-cllaw 48677

This theorem is referenced by: (None)