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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 6435 | . 2 ⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦)) | |
2 | vex 2752 | . . 3 ⊢ 𝑦 ∈ V | |
3 | vex 2752 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | oafnex 6459 | . . . 4 ⊢ (𝑧 ∈ V ↦ suc 𝑧) Fn V | |
5 | 3, 4 | rdgexg 6404 | . . 3 ⊢ (𝑦 ∈ V → (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦) ∈ V) |
6 | 2, 5 | ax-mp 5 | . 2 ⊢ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦) ∈ V |
7 | 1, 6 | fnmpoi 6219 | 1 ⊢ +o Fn (On × On) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2158 Vcvv 2749 ↦ cmpt 4076 Oncon0 4375 suc csuc 4377 × cxp 4636 Fn wfn 5223 ‘cfv 5228 reccrdg 6384 +o coa 6428 |
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-in1 615 ax-in2 616 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-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-recs 6320 df-irdg 6385 df-oadd 6435 |
This theorem is referenced by: dmaddpi 7338 |
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