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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 | โข โo Fn (On ร On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-oexp 8471 | . 2 โข โo = (๐ฅ โ On, ๐ฆ โ On โฆ if(๐ฅ = โ , (1o โ ๐ฆ), (rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o)โ๐ฆ))) | |
2 | 1on 8477 | . . . 4 โข 1o โ On | |
3 | difexg 5327 | . . . 4 โข (1o โ On โ (1o โ ๐ฆ) โ V) | |
4 | 2, 3 | ax-mp 5 | . . 3 โข (1o โ ๐ฆ) โ V |
5 | fvex 6904 | . . 3 โข (rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o)โ๐ฆ) โ V | |
6 | 4, 5 | ifex 4578 | . 2 โข if(๐ฅ = โ , (1o โ ๐ฆ), (rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o)โ๐ฆ)) โ V |
7 | 1, 6 | fnmpoi 8055 | 1 โข โo Fn (On ร On) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 โ wcel 2106 Vcvv 3474 โ cdif 3945 โ c0 4322 ifcif 4528 โฆ cmpt 5231 ร cxp 5674 Oncon0 6364 Fn wfn 6538 โcfv 6543 (class class class)co 7408 reccrdg 8408 1oc1o 8458 ยทo comu 8463 โo coe 8464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-1o 8465 df-oexp 8471 |
This theorem is referenced by: oaabs2 8647 omabs 8649 |
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