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Mirrors > Home > ILE Home > Th. List > fnoa | GIF version |
Description: Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.) |
Ref | Expression |
---|---|
fnoa | ⊢ +o Fn (On × On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-oadd 6247 | . 2 ⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦)) | |
2 | vex 2644 | . . 3 ⊢ 𝑦 ∈ V | |
3 | vex 2644 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | oafnex 6270 | . . . 4 ⊢ (𝑧 ∈ V ↦ suc 𝑧) Fn V | |
5 | 3, 4 | rdgexg 6216 | . . 3 ⊢ (𝑦 ∈ V → (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦) ∈ V) |
6 | 2, 5 | ax-mp 7 | . 2 ⊢ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦) ∈ V |
7 | 1, 6 | fnmpoi 6032 | 1 ⊢ +o Fn (On × On) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1448 Vcvv 2641 ↦ cmpt 3929 Oncon0 4223 suc csuc 4225 × cxp 4475 Fn wfn 5054 ‘cfv 5059 reccrdg 6196 +o coa 6240 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-iord 4226 df-on 4228 df-suc 4231 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-irdg 6197 df-oadd 6247 |
This theorem is referenced by: dmaddpi 7034 |
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