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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 6419 | . 2 ⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)) | |
2 | vex 2740 | . . 3 ⊢ 𝑦 ∈ V | |
3 | 1on 6420 | . . . . 5 ⊢ 1o ∈ On | |
4 | 3 | elexi 2749 | . . . 4 ⊢ 1o ∈ V |
5 | vex 2740 | . . . . . 6 ⊢ 𝑧 ∈ V | |
6 | vex 2740 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | omexg 6448 | . . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝑥 ∈ V) → (𝑧 ·o 𝑥) ∈ V) | |
8 | 5, 6, 7 | mp2an 426 | . . . . 5 ⊢ (𝑧 ·o 𝑥) ∈ V |
9 | eqid 2177 | . . . . 5 ⊢ (𝑧 ∈ V ↦ (𝑧 ·o 𝑥)) = (𝑧 ∈ V ↦ (𝑧 ·o 𝑥)) | |
10 | 8, 9 | fnmpti 5342 | . . . 4 ⊢ (𝑧 ∈ V ↦ (𝑧 ·o 𝑥)) Fn V |
11 | 4, 10 | rdgexg 6386 | . . 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 6201 | 1 ⊢ ↑o Fn (On × On) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 Vcvv 2737 ↦ cmpt 4063 Oncon0 4362 × cxp 4623 Fn wfn 5209 ‘cfv 5214 (class class class)co 5871 reccrdg 6366 1oc1o 6406 ·o comu 6411 ↑o coei 6412 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-iord 4365 df-on 4367 df-suc 4370 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5176 df-fun 5216 df-fn 5217 df-f 5218 df-f1 5219 df-fo 5220 df-f1o 5221 df-fv 5222 df-ov 5874 df-oprab 5875 df-mpo 5876 df-1st 6137 df-2nd 6138 df-recs 6302 df-irdg 6367 df-1o 6413 df-oadd 6417 df-omul 6418 df-oexpi 6419 |
This theorem is referenced by: (None) |
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