Theorem caovcld 7593

Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovclg.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)
caovcld.2	⊢ (𝜑 → 𝐴 ∈ 𝐶)
caovcld.3	⊢ (𝜑 → 𝐵 ∈ 𝐷)

Assertion

Ref	Expression
caovcld	⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐸,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem caovcld

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (𝜑 → 𝜑)
2		caovcld.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
3		caovcld.3	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
4		caovclg.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)
5	4	caovclg 7592	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)
6	1, 2, 3, 5	syl12anc 849	1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145  (class class class)co 7400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-ov 7403

This theorem is referenced by:  caovdir2d  7616  caov4d  7624  climcn2  15634  grpinva  18722  plydivlem1  26415  plydivlem4  26418