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Mirrors > Home > ILE Home > Th. List > rdgivallem | Unicode version |
Description: Value of the recursive definition generator. Lemma for rdgival 5969 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rdgivallem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-irdg 5957 | . . . 4 recs | |
2 | rdgruledefgg 5962 | . . . . 5 | |
3 | 2 | alrimiv 1754 | . . . 4 |
4 | 1, 3 | tfri2d 5950 | . . 3 |
5 | 4 | 3impa 1099 | . 2 |
6 | eqidd 2041 | . . 3 | |
7 | dmeq 4535 | . . . . . 6 | |
8 | onss 4219 | . . . . . . . . 9 | |
9 | 8 | 3ad2ant3 927 | . . . . . . . 8 |
10 | rdgifnon 5966 | . . . . . . . . . 10 | |
11 | fndm 4998 | . . . . . . . . . 10 | |
12 | 10, 11 | syl 14 | . . . . . . . . 9 |
13 | 12 | 3adant3 924 | . . . . . . . 8 |
14 | 9, 13 | sseqtr4d 2982 | . . . . . . 7 |
15 | ssdmres 4633 | . . . . . . 7 | |
16 | 14, 15 | sylib 127 | . . . . . 6 |
17 | 7, 16 | sylan9eqr 2094 | . . . . 5 |
18 | fveq1 5177 | . . . . . . 7 | |
19 | 18 | fveq2d 5182 | . . . . . 6 |
20 | 19 | adantl 262 | . . . . 5 |
21 | 17, 20 | iuneq12d 3681 | . . . 4 |
22 | 21 | uneq2d 3097 | . . 3 |
23 | rdgfun 5960 | . . . . 5 | |
24 | resfunexg 5382 | . . . . 5 | |
25 | 23, 24 | mpan 400 | . . . 4 |
26 | 25 | 3ad2ant3 927 | . . 3 |
27 | simpr 103 | . . . . . 6 | |
28 | vex 2560 | . . . . . . . . . 10 | |
29 | fvexg 5194 | . . . . . . . . . 10 | |
30 | 25, 28, 29 | sylancl 392 | . . . . . . . . 9 |
31 | 30 | ralrimivw 2393 | . . . . . . . 8 |
32 | 31 | adantl 262 | . . . . . . 7 |
33 | funfvex 5192 | . . . . . . . . . . 11 | |
34 | 33 | funfni 4999 | . . . . . . . . . 10 |
35 | 34 | ex 108 | . . . . . . . . 9 |
36 | 35 | ralimdv 2388 | . . . . . . . 8 |
37 | 36 | adantr 261 | . . . . . . 7 |
38 | 32, 37 | mpd 13 | . . . . . 6 |
39 | iunexg 5746 | . . . . . 6 | |
40 | 27, 38, 39 | syl2anc 391 | . . . . 5 |
41 | 40 | 3adant2 923 | . . . 4 |
42 | unexg 4178 | . . . . . 6 | |
43 | 42 | ex 108 | . . . . 5 |
44 | 43 | 3ad2ant2 926 | . . . 4 |
45 | 41, 44 | mpd 13 | . . 3 |
46 | 6, 22, 26, 45 | fvmptd 5253 | . 2 |
47 | 5, 46 | eqtrd 2072 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 wral 2306 cvv 2557 cun 2915 wss 2917 ciun 3657 cmpt 3818 con0 4100 cdm 4345 cres 4347 wfun 4896 wfn 4897 cfv 4902 crdg 5956 |
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-in1 544 ax-in2 545 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 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 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-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 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-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 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 df-recs 5920 df-irdg 5957 |
This theorem is referenced by: rdgival 5969 rdgon 5973 |
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