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Mirrors > Home > ILE Home > Th. List > caovcl | 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 1367 | . 2 ⊢ ⊤ | |
2 | caovcl.1 | . . . 4 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) | |
3 | 2 | adantl 277 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
4 | 3 | caovclg 6041 | . 2 ⊢ ((⊤ ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝑆) |
5 | 1, 4 | mpan 424 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ⊤wtru 1364 ∈ wcel 2158 (class class class)co 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-iota 5190 df-fv 5236 df-ov 5891 |
This theorem is referenced by: ecopovtrn 6646 ecopovtrng 6649 genpelvl 7525 genpelvu 7526 genpml 7530 genpmu 7531 genprndl 7534 genprndu 7535 genpassl 7537 genpassu 7538 genpassg 7539 expcllem 10545 |
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