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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 8422 | . 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 8428 | . . . 4 โข 1o โ On | |
3 | difexg 5288 | . . . 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 6859 | . . 3 โข (rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o)โ๐ฆ) โ V | |
6 | 4, 5 | ifex 4540 | . 2 โข if(๐ฅ = โ , (1o โ ๐ฆ), (rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o)โ๐ฆ)) โ V |
7 | 1, 6 | fnmpoi 8006 | 1 โข โo Fn (On ร On) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 Vcvv 3447 โ cdif 3911 โ c0 4286 ifcif 4490 โฆ cmpt 5192 ร cxp 5635 Oncon0 6321 Fn wfn 6495 โcfv 6500 (class class class)co 7361 reccrdg 8359 1oc1o 8409 ยทo comu 8414 โo coe 8415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-1o 8416 df-oexp 8422 |
This theorem is referenced by: oaabs2 8599 omabs 8601 |
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