Theorem caoprcl 3066

Description: Convert an operation closure law to class notation.

Hypothesis

Ref	Expression
caoprcl.1	⊢ ((x ∈ S ∧ y ∈ S) → (xFy) ∈ S)

Assertion

Ref	Expression
caoprcl	⊢ ((A ∈ S ∧ B ∈ S) → (AFB) ∈ S)

Distinct variable group(s):   x,y,F   x,S,y   x,A,y   x,B,y

Proof of Theorem caoprcl

Step	Hyp	Ref	Expression
1		opreq1 3006	. . 3 ⊢ (x = A → (xFy) = (AFy))
2	1	eleq1d 1155	. 2 ⊢ (x = A → ((xFy) ∈ S ↔ (AFy) ∈ S))
3		opreq2 3007	. . 3 ⊢ (y = B → (AFy) = (AFB))
4	3	eleq1d 1155	. 2 ⊢ (y = B → ((AFy) ∈ S ↔ (AFB) ∈ S))
5		caoprcl.1	. 2 ⊢ ((x ∈ S ∧ y ∈ S) → (xFy) ∈ S)
6	2, 4, 5	vtocl2ga 1388	1 ⊢ ((A ∈ S ∧ B ∈ S) → (AFB) ∈ S)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  (class class class)co 3001

This theorem is referenced by:  ecopoprtrn 3247  eceqopreq 3249  genpcl 3905  genpass 3906

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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