![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > fnom | GIF version |
Description: Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.) |
Ref | Expression |
---|---|
fnom | โข ยทo Fn (On ร On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-omul 6436 | . 2 โข ยทo = (๐ฅ โ On, ๐ฆ โ On โฆ (rec((๐ง โ V โฆ (๐ง +o ๐ฅ)), โ )โ๐ฆ)) | |
2 | vex 2752 | . . 3 โข ๐ฆ โ V | |
3 | 0ex 4142 | . . . 4 โข โ โ V | |
4 | vex 2752 | . . . . 5 โข ๐ฅ โ V | |
5 | omfnex 6464 | . . . . 5 โข (๐ฅ โ V โ (๐ง โ V โฆ (๐ง +o ๐ฅ)) Fn V) | |
6 | 4, 5 | ax-mp 5 | . . . 4 โข (๐ง โ V โฆ (๐ง +o ๐ฅ)) Fn V |
7 | 3, 6 | rdgexg 6404 | . . 3 โข (๐ฆ โ V โ (rec((๐ง โ V โฆ (๐ง +o ๐ฅ)), โ )โ๐ฆ) โ V) |
8 | 2, 7 | ax-mp 5 | . 2 โข (rec((๐ง โ V โฆ (๐ง +o ๐ฅ)), โ )โ๐ฆ) โ V |
9 | 1, 8 | fnmpoi 6219 | 1 โข ยทo Fn (On ร On) |
Colors of variables: wff set class |
Syntax hints: โ wcel 2158 Vcvv 2749 โ c0 3434 โฆ cmpt 4076 Oncon0 4375 ร cxp 4636 Fn wfn 5223 โcfv 5228 (class class class)co 5888 reccrdg 6384 +o coa 6428 ยทo comu 6429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-recs 6320 df-irdg 6385 df-oadd 6435 df-omul 6436 |
This theorem is referenced by: dmmulpi 7339 |
Copyright terms: Public domain | W3C validator |