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Mirrors > Home > ILE Home > Th. List > rdgivallem | Unicode version |
Description: Value of the recursive definition generator. Lemma for rdgival 6247 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 6235 | . . . 4 recs | |
2 | rdgruledefgg 6240 | . . . . 5 | |
3 | 2 | alrimiv 1830 | . . . 4 |
4 | 1, 3 | tfri2d 6201 | . . 3 |
5 | 4 | 3impa 1161 | . 2 |
6 | eqidd 2118 | . . 3 | |
7 | dmeq 4709 | . . . . . 6 | |
8 | onss 4379 | . . . . . . . . 9 | |
9 | 8 | 3ad2ant3 989 | . . . . . . . 8 |
10 | rdgifnon 6244 | . . . . . . . . . 10 | |
11 | fndm 5192 | . . . . . . . . . 10 | |
12 | 10, 11 | syl 14 | . . . . . . . . 9 |
13 | 12 | 3adant3 986 | . . . . . . . 8 |
14 | 9, 13 | sseqtrrd 3106 | . . . . . . 7 |
15 | ssdmres 4811 | . . . . . . 7 | |
16 | 14, 15 | sylib 121 | . . . . . 6 |
17 | 7, 16 | sylan9eqr 2172 | . . . . 5 |
18 | fveq1 5388 | . . . . . . 7 | |
19 | 18 | fveq2d 5393 | . . . . . 6 |
20 | 19 | adantl 275 | . . . . 5 |
21 | 17, 20 | iuneq12d 3807 | . . . 4 |
22 | 21 | uneq2d 3200 | . . 3 |
23 | rdgfun 6238 | . . . . 5 | |
24 | resfunexg 5609 | . . . . 5 | |
25 | 23, 24 | mpan 420 | . . . 4 |
26 | 25 | 3ad2ant3 989 | . . 3 |
27 | simpr 109 | . . . . . 6 | |
28 | vex 2663 | . . . . . . . . . 10 | |
29 | fvexg 5408 | . . . . . . . . . 10 | |
30 | 25, 28, 29 | sylancl 409 | . . . . . . . . 9 |
31 | 30 | ralrimivw 2483 | . . . . . . . 8 |
32 | 31 | adantl 275 | . . . . . . 7 |
33 | funfvex 5406 | . . . . . . . . . . 11 | |
34 | 33 | funfni 5193 | . . . . . . . . . 10 |
35 | 34 | ex 114 | . . . . . . . . 9 |
36 | 35 | ralimdv 2477 | . . . . . . . 8 |
37 | 36 | adantr 274 | . . . . . . 7 |
38 | 32, 37 | mpd 13 | . . . . . 6 |
39 | iunexg 5985 | . . . . . 6 | |
40 | 27, 38, 39 | syl2anc 408 | . . . . 5 |
41 | 40 | 3adant2 985 | . . . 4 |
42 | unexg 4334 | . . . . . 6 | |
43 | 42 | ex 114 | . . . . 5 |
44 | 43 | 3ad2ant2 988 | . . . 4 |
45 | 41, 44 | mpd 13 | . . 3 |
46 | 6, 22, 26, 45 | fvmptd 5470 | . 2 |
47 | 5, 46 | eqtrd 2150 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 wral 2393 cvv 2660 cun 3039 wss 3041 ciun 3783 cmpt 3959 con0 4255 cdm 4509 cres 4511 wfun 5087 wfn 5088 cfv 5093 crdg 6234 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-recs 6170 df-irdg 6235 |
This theorem is referenced by: rdgival 6247 |
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