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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 6399 | . 2 ⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦)) | |
2 | vex 2733 | . . 3 ⊢ 𝑦 ∈ V | |
3 | vex 2733 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | oafnex 6423 | . . . 4 ⊢ (𝑧 ∈ V ↦ suc 𝑧) Fn V | |
5 | 3, 4 | rdgexg 6368 | . . 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 6183 | 1 ⊢ +o Fn (On × On) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 Vcvv 2730 ↦ cmpt 4050 Oncon0 4348 suc csuc 4350 × cxp 4609 Fn wfn 5193 ‘cfv 5198 reccrdg 6348 +o coa 6392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 |
This theorem is referenced by: dmaddpi 7287 |
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