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Mirrors > Home > ILE Home > Th. List > rdgruledefgg | Unicode version |
Description: The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Ref | Expression |
---|---|
rdgruledefgg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2566 | . 2 | |
2 | funmpt 4938 | . . . 4 | |
3 | vex 2560 | . . . . 5 | |
4 | vex 2560 | . . . . . . . . . . . . 13 | |
5 | vex 2560 | . . . . . . . . . . . . 13 | |
6 | 4, 5 | fvex 5195 | . . . . . . . . . . . 12 |
7 | funfvex 5192 | . . . . . . . . . . . . 13 | |
8 | 7 | funfni 4999 | . . . . . . . . . . . 12 |
9 | 6, 8 | mpan2 401 | . . . . . . . . . . 11 |
10 | 9 | ralrimivw 2393 | . . . . . . . . . 10 |
11 | 4 | dmex 4598 | . . . . . . . . . . 11 |
12 | iunexg 5746 | . . . . . . . . . . 11 | |
13 | 11, 12 | mpan 400 | . . . . . . . . . 10 |
14 | 10, 13 | syl 14 | . . . . . . . . 9 |
15 | unexg 4178 | . . . . . . . . 9 | |
16 | 14, 15 | sylan2 270 | . . . . . . . 8 |
17 | 16 | ancoms 255 | . . . . . . 7 |
18 | 17 | ralrimivw 2393 | . . . . . 6 |
19 | dmmptg 4818 | . . . . . 6 | |
20 | 18, 19 | syl 14 | . . . . 5 |
21 | 3, 20 | syl5eleqr 2127 | . . . 4 |
22 | funfvex 5192 | . . . 4 | |
23 | 2, 21, 22 | sylancr 393 | . . 3 |
24 | 23, 2 | jctil 295 | . 2 |
25 | 1, 24 | sylan2 270 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 wral 2306 cvv 2557 cun 2915 ciun 3657 cmpt 3818 cdm 4345 wfun 4896 wfn 4897 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 |
This theorem is referenced by: rdgruledefg 5963 rdgexggg 5964 rdgifnon 5966 rdgivallem 5968 |
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