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Mirrors > Home > ILE Home > Th. List > fnoei | GIF version |
Description: Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.) |
Ref | Expression |
---|---|
fnoei | ⊢ ↑o Fn (On × On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-oexpi 6312 | . 2 ⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)) | |
2 | vex 2684 | . . 3 ⊢ 𝑦 ∈ V | |
3 | 1on 6313 | . . . . 5 ⊢ 1o ∈ On | |
4 | 3 | elexi 2693 | . . . 4 ⊢ 1o ∈ V |
5 | vex 2684 | . . . . . 6 ⊢ 𝑧 ∈ V | |
6 | vex 2684 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | omexg 6340 | . . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝑥 ∈ V) → (𝑧 ·o 𝑥) ∈ V) | |
8 | 5, 6, 7 | mp2an 422 | . . . . 5 ⊢ (𝑧 ·o 𝑥) ∈ V |
9 | eqid 2137 | . . . . 5 ⊢ (𝑧 ∈ V ↦ (𝑧 ·o 𝑥)) = (𝑧 ∈ V ↦ (𝑧 ·o 𝑥)) | |
10 | 8, 9 | fnmpti 5246 | . . . 4 ⊢ (𝑧 ∈ V ↦ (𝑧 ·o 𝑥)) Fn V |
11 | 4, 10 | rdgexg 6279 | . . 3 ⊢ (𝑦 ∈ V → (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦) ∈ V) |
12 | 2, 11 | ax-mp 5 | . 2 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦) ∈ V |
13 | 1, 12 | fnmpoi 6095 | 1 ⊢ ↑o Fn (On × On) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2681 ↦ cmpt 3984 Oncon0 4280 × cxp 4532 Fn wfn 5113 ‘cfv 5118 (class class class)co 5767 reccrdg 6259 1oc1o 6299 ·o comu 6304 ↑o coei 6305 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-oadd 6310 df-omul 6311 df-oexpi 6312 |
This theorem is referenced by: (None) |
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