Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2610 | . 2 | |
2 | funmpt 4958 | . . . 4 | |
3 | vex 2604 | . . . . 5 | |
4 | vex 2604 | . . . . . . . . . . . . 13 | |
5 | vex 2604 | . . . . . . . . . . . . 13 | |
6 | 4, 5 | fvex 5215 | . . . . . . . . . . . 12 |
7 | funfvex 5212 | . . . . . . . . . . . . 13 | |
8 | 7 | funfni 5019 | . . . . . . . . . . . 12 |
9 | 6, 8 | mpan2 415 | . . . . . . . . . . 11 |
10 | 9 | ralrimivw 2435 | . . . . . . . . . 10 |
11 | 4 | dmex 4616 | . . . . . . . . . . 11 |
12 | iunexg 5766 | . . . . . . . . . . 11 | |
13 | 11, 12 | mpan 414 | . . . . . . . . . 10 |
14 | 10, 13 | syl 14 | . . . . . . . . 9 |
15 | unexg 4196 | . . . . . . . . 9 | |
16 | 14, 15 | sylan2 280 | . . . . . . . 8 |
17 | 16 | ancoms 264 | . . . . . . 7 |
18 | 17 | ralrimivw 2435 | . . . . . 6 |
19 | dmmptg 4838 | . . . . . 6 | |
20 | 18, 19 | syl 14 | . . . . 5 |
21 | 3, 20 | syl5eleqr 2168 | . . . 4 |
22 | funfvex 5212 | . . . 4 | |
23 | 2, 21, 22 | sylancr 405 | . . 3 |
24 | 23, 2 | jctil 305 | . 2 |
25 | 1, 24 | sylan2 280 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 wral 2348 cvv 2601 cun 2971 ciun 3678 cmpt 3839 cdm 4363 wfun 4916 wfn 4917 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 |
This theorem is referenced by: rdgruledefg 5986 rdgexggg 5987 rdgifnon 5989 rdgivallem 5991 |
Copyright terms: Public domain | W3C validator |