Theorem caovcld 5674

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 19	. 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 5673	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)
6	1, 2, 3, 5	syl12anc 1167	1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)

Syntax hints:   → wi 4   ∧ wa 102   ∈ wcel 1433  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  caovdir2d  5697  caov4d  5705  caovdilemd  5712  caovlem2d  5713  grprinvd  5716  ecopovtrn  6226  ecopovtrng  6229  ordpipqqs  6564  ltanqg  6590  ltmnqg  6591  recexprlem1ssu  6824  mulgt0sr  6954  mulextsr1lem  6956  axmulass  7039  frec2uzrdg  9411  frecuzrdgsuc  9417  iseqovex  9439  iseqval  9440  iseqp1  9445  iseqdistr  9470  climcn2  10148