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Mirrors > Home > ILE Home > Th. List > oafnex | GIF version |
Description: The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.) |
Ref | Expression |
---|---|
oafnex | ⊢ (𝑥 ∈ V ↦ suc 𝑥) Fn V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2663 | . . 3 ⊢ 𝑥 ∈ V | |
2 | 1 | sucex 4385 | . 2 ⊢ suc 𝑥 ∈ V |
3 | eqid 2117 | . 2 ⊢ (𝑥 ∈ V ↦ suc 𝑥) = (𝑥 ∈ V ↦ suc 𝑥) | |
4 | 2, 3 | fnmpti 5221 | 1 ⊢ (𝑥 ∈ V ↦ suc 𝑥) Fn V |
Colors of variables: wff set class |
Syntax hints: Vcvv 2660 ↦ cmpt 3959 suc csuc 4257 Fn wfn 5088 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-fun 5095 df-fn 5096 |
This theorem is referenced by: fnoa 6311 oaexg 6312 oav 6318 oav2 6327 oawordi 6333 |
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