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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 8468 | . 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 8474 | . . . 4 โข 1o โ On | |
3 | difexg 5318 | . . . 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 6895 | . . 3 โข (rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o)โ๐ฆ) โ V | |
6 | 4, 5 | ifex 4571 | . 2 โข if(๐ฅ = โ , (1o โ ๐ฆ), (rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o)โ๐ฆ)) โ V |
7 | 1, 6 | fnmpoi 8050 | 1 โข โo Fn (On ร On) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 โ wcel 2098 Vcvv 3466 โ cdif 3938 โ c0 4315 ifcif 4521 โฆ cmpt 5222 ร cxp 5665 Oncon0 6355 Fn wfn 6529 โcfv 6534 (class class class)co 7402 reccrdg 8405 1oc1o 8455 ยทo comu 8460 โo coe 8461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ord 6358 df-on 6359 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-1o 8462 df-oexp 8468 |
This theorem is referenced by: oaabs2 8645 omabs 8647 |
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