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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 7453 | . 2 ⊢ ↑𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1𝑜 ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦))) | |
2 | 1on 7454 | . . . 4 ⊢ 1𝑜 ∈ On | |
3 | difexg 4735 | . . . 4 ⊢ (1𝑜 ∈ On → (1𝑜 ∖ 𝑦) ∈ V) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (1𝑜 ∖ 𝑦) ∈ V |
5 | fvex 6113 | . . 3 ⊢ (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦) ∈ V | |
6 | 4, 5 | ifex 4106 | . 2 ⊢ if(𝑥 = ∅, (1𝑜 ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)) ∈ V |
7 | 1, 6 | fnmpt2i 7128 | 1 ⊢ ↑𝑜 Fn (On × On) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 ifcif 4036 ↦ cmpt 4643 × cxp 5036 Oncon0 5640 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 reccrdg 7392 1𝑜c1o 7440 ·𝑜 comu 7445 ↑𝑜 coe 7446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-1o 7447 df-oexp 7453 |
This theorem is referenced by: oaabs2 7612 omabs 7614 |
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