Theorem caovcl 5655

Description: Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)

Hypothesis

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

Assertion

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

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem caovcl

Step	Hyp	Ref	Expression
1		tru 1247	. 2 ⊢ ⊤
2		caovcl.1	. . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆)
3	2	adantl 262	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
4	3	caovclg 5653	. 2 ⊢ ((⊤ ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝑆)
5	1, 4	mpan 400	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 97  ⊤wtru 1244   ∈ wcel 1393  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  ecopovtrn  6203  ecopovtrng  6206  genpelvl  6610  genpelvu  6611  genpml  6615  genpmu  6616  genprndl  6619  genprndu  6620  genpassl  6622  genpassu  6623  genpassg  6624  expcllem  9266