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Mirrors > Home > ILE Home > Th. List > rdgivallem | Unicode version |
Description: Value of the recursive definition generator. Lemma for rdgival 6279 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 6267 | . . . 4 recs | |
2 | rdgruledefgg 6272 | . . . . 5 | |
3 | 2 | alrimiv 1846 | . . . 4 |
4 | 1, 3 | tfri2d 6233 | . . 3 |
5 | 4 | 3impa 1176 | . 2 |
6 | eqidd 2140 | . . 3 | |
7 | dmeq 4739 | . . . . . 6 | |
8 | onss 4409 | . . . . . . . . 9 | |
9 | 8 | 3ad2ant3 1004 | . . . . . . . 8 |
10 | rdgifnon 6276 | . . . . . . . . . 10 | |
11 | fndm 5222 | . . . . . . . . . 10 | |
12 | 10, 11 | syl 14 | . . . . . . . . 9 |
13 | 12 | 3adant3 1001 | . . . . . . . 8 |
14 | 9, 13 | sseqtrrd 3136 | . . . . . . 7 |
15 | ssdmres 4841 | . . . . . . 7 | |
16 | 14, 15 | sylib 121 | . . . . . 6 |
17 | 7, 16 | sylan9eqr 2194 | . . . . 5 |
18 | fveq1 5420 | . . . . . . 7 | |
19 | 18 | fveq2d 5425 | . . . . . 6 |
20 | 19 | adantl 275 | . . . . 5 |
21 | 17, 20 | iuneq12d 3837 | . . . 4 |
22 | 21 | uneq2d 3230 | . . 3 |
23 | rdgfun 6270 | . . . . 5 | |
24 | resfunexg 5641 | . . . . 5 | |
25 | 23, 24 | mpan 420 | . . . 4 |
26 | 25 | 3ad2ant3 1004 | . . 3 |
27 | simpr 109 | . . . . . 6 | |
28 | vex 2689 | . . . . . . . . . 10 | |
29 | fvexg 5440 | . . . . . . . . . 10 | |
30 | 25, 28, 29 | sylancl 409 | . . . . . . . . 9 |
31 | 30 | ralrimivw 2506 | . . . . . . . 8 |
32 | 31 | adantl 275 | . . . . . . 7 |
33 | funfvex 5438 | . . . . . . . . . . 11 | |
34 | 33 | funfni 5223 | . . . . . . . . . 10 |
35 | 34 | ex 114 | . . . . . . . . 9 |
36 | 35 | ralimdv 2500 | . . . . . . . 8 |
37 | 36 | adantr 274 | . . . . . . 7 |
38 | 32, 37 | mpd 13 | . . . . . 6 |
39 | iunexg 6017 | . . . . . 6 | |
40 | 27, 38, 39 | syl2anc 408 | . . . . 5 |
41 | 40 | 3adant2 1000 | . . . 4 |
42 | unexg 4364 | . . . . . 6 | |
43 | 42 | ex 114 | . . . . 5 |
44 | 43 | 3ad2ant2 1003 | . . . 4 |
45 | 41, 44 | mpd 13 | . . 3 |
46 | 6, 22, 26, 45 | fvmptd 5502 | . 2 |
47 | 5, 46 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wral 2416 cvv 2686 cun 3069 wss 3071 ciun 3813 cmpt 3989 con0 4285 cdm 4539 cres 4541 wfun 5117 wfn 5118 cfv 5123 crdg 6266 |
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 603 ax-in2 604 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 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-recs 6202 df-irdg 6267 |
This theorem is referenced by: rdgival 6279 |
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