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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 2692 | . 2 | |
2 | funmpt 5156 | . . . 4 | |
3 | vex 2684 | . . . . 5 | |
4 | vex 2684 | . . . . . . . . . . . . 13 | |
5 | vex 2684 | . . . . . . . . . . . . 13 | |
6 | 4, 5 | fvex 5434 | . . . . . . . . . . . 12 |
7 | funfvex 5431 | . . . . . . . . . . . . 13 | |
8 | 7 | funfni 5218 | . . . . . . . . . . . 12 |
9 | 6, 8 | mpan2 421 | . . . . . . . . . . 11 |
10 | 9 | ralrimivw 2504 | . . . . . . . . . 10 |
11 | 4 | dmex 4800 | . . . . . . . . . . 11 |
12 | iunexg 6010 | . . . . . . . . . . 11 | |
13 | 11, 12 | mpan 420 | . . . . . . . . . 10 |
14 | 10, 13 | syl 14 | . . . . . . . . 9 |
15 | unexg 4359 | . . . . . . . . 9 | |
16 | 14, 15 | sylan2 284 | . . . . . . . 8 |
17 | 16 | ancoms 266 | . . . . . . 7 |
18 | 17 | ralrimivw 2504 | . . . . . 6 |
19 | dmmptg 5031 | . . . . . 6 | |
20 | 18, 19 | syl 14 | . . . . 5 |
21 | 3, 20 | eleqtrrid 2227 | . . . 4 |
22 | funfvex 5431 | . . . 4 | |
23 | 2, 21, 22 | sylancr 410 | . . 3 |
24 | 23, 2 | jctil 310 | . 2 |
25 | 1, 24 | sylan2 284 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2414 cvv 2681 cun 3064 ciun 3808 cmpt 3984 cdm 4534 wfun 5112 wfn 5113 cfv 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 |
This theorem is referenced by: rdgruledefg 6266 rdgexggg 6267 rdgifnon 6269 rdgivallem 6271 |
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