Theorem caovcld 7644

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 7643	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)
6	1, 2, 3, 5	syl12anc 850	1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  (class class class)co 7449

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6518  df-fv 6572  df-ov 7452

This theorem is used by:  caovdir2d  7667  caov4d  7675  climcn2  15801  grpinva  18896  plydivlem1  26658  plydivlem4  26661