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| Mirrors > Home > MPE Home > Th. List > fnoe | Structured version Visualization version GIF version | ||
| Description: Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) |
| Ref | Expression |
|---|---|
| fnoe | ⊢ ↑o Fn (On × On) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-oexp 8512 | . 2 ⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))) | |
| 2 | 1on 8518 | . . . 4 ⊢ 1o ∈ On | |
| 3 | difexg 5329 | . . . 4 ⊢ (1o ∈ On → (1o ∖ 𝑦) ∈ V) | |
| 4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (1o ∖ 𝑦) ∈ V |
| 5 | fvex 6919 | . . 3 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦) ∈ V | |
| 6 | 4, 5 | ifex 4576 | . 2 ⊢ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)) ∈ V |
| 7 | 1, 6 | fnmpoi 8095 | 1 ⊢ ↑o Fn (On × On) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 ifcif 4525 ↦ cmpt 5225 × cxp 5683 Oncon0 6384 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 reccrdg 8449 1oc1o 8499 ·o comu 8504 ↑o coe 8505 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-1o 8506 df-oexp 8512 |
| This theorem is referenced by: oaabs2 8687 omabs 8689 |
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