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Mirrors > Home > MPE Home > Th. List > caovcl | Structured version Visualization version GIF version |
Description: Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.) |
Ref | Expression |
---|---|
caovcl.1 | ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
caovcl | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1547 | . 2 ⊢ ⊤ | |
2 | caovcl.1 | . . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) | |
3 | 2 | adantl 485 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
4 | 3 | caovclg 7378 | . 2 ⊢ ((⊤ ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝑆) |
5 | 1, 4 | mpan 690 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ⊤wtru 1544 ∈ wcel 2112 (class class class)co 7191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 |
This theorem is referenced by: ecopovtrn 8480 eceqoveq 8482 genpss 10583 genpnnp 10584 genpass 10588 expcllem 13611 txlly 22487 txnlly 22488 |
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