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Mirrors > Home > ILE Home > Th. List > rdgivallem | Unicode version |
Description: Value of the recursive definition generator. Lemma for rdgival 6373 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 6361 | . . . 4 recs | |
2 | rdgruledefgg 6366 | . . . . 5 | |
3 | 2 | alrimiv 1872 | . . . 4 |
4 | 1, 3 | tfri2d 6327 | . . 3 |
5 | 4 | 3impa 1194 | . 2 |
6 | eqidd 2176 | . . 3 | |
7 | dmeq 4820 | . . . . . 6 | |
8 | onss 4486 | . . . . . . . . 9 | |
9 | 8 | 3ad2ant3 1020 | . . . . . . . 8 |
10 | rdgifnon 6370 | . . . . . . . . . 10 | |
11 | fndm 5307 | . . . . . . . . . 10 | |
12 | 10, 11 | syl 14 | . . . . . . . . 9 |
13 | 12 | 3adant3 1017 | . . . . . . . 8 |
14 | 9, 13 | sseqtrrd 3192 | . . . . . . 7 |
15 | ssdmres 4922 | . . . . . . 7 | |
16 | 14, 15 | sylib 122 | . . . . . 6 |
17 | 7, 16 | sylan9eqr 2230 | . . . . 5 |
18 | fveq1 5506 | . . . . . . 7 | |
19 | 18 | fveq2d 5511 | . . . . . 6 |
20 | 19 | adantl 277 | . . . . 5 |
21 | 17, 20 | iuneq12d 3906 | . . . 4 |
22 | 21 | uneq2d 3287 | . . 3 |
23 | rdgfun 6364 | . . . . 5 | |
24 | resfunexg 5729 | . . . . 5 | |
25 | 23, 24 | mpan 424 | . . . 4 |
26 | 25 | 3ad2ant3 1020 | . . 3 |
27 | simpr 110 | . . . . . 6 | |
28 | vex 2738 | . . . . . . . . . 10 | |
29 | fvexg 5526 | . . . . . . . . . 10 | |
30 | 25, 28, 29 | sylancl 413 | . . . . . . . . 9 |
31 | 30 | ralrimivw 2549 | . . . . . . . 8 |
32 | 31 | adantl 277 | . . . . . . 7 |
33 | funfvex 5524 | . . . . . . . . . . 11 | |
34 | 33 | funfni 5308 | . . . . . . . . . 10 |
35 | 34 | ex 115 | . . . . . . . . 9 |
36 | 35 | ralimdv 2543 | . . . . . . . 8 |
37 | 36 | adantr 276 | . . . . . . 7 |
38 | 32, 37 | mpd 13 | . . . . . 6 |
39 | iunexg 6110 | . . . . . 6 | |
40 | 27, 38, 39 | syl2anc 411 | . . . . 5 |
41 | 40 | 3adant2 1016 | . . . 4 |
42 | unexg 4437 | . . . . . 6 | |
43 | 42 | ex 115 | . . . . 5 |
44 | 43 | 3ad2ant2 1019 | . . . 4 |
45 | 41, 44 | mpd 13 | . . 3 |
46 | 6, 22, 26, 45 | fvmptd 5589 | . 2 |
47 | 5, 46 | eqtrd 2208 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 w3a 978 wceq 1353 wcel 2146 wral 2453 cvv 2735 cun 3125 wss 3127 ciun 3882 cmpt 4059 con0 4357 cdm 4620 cres 4622 wfun 5202 wfn 5203 cfv 5208 crdg 6360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-suc 4365 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-recs 6296 df-irdg 6361 |
This theorem is referenced by: rdgival 6373 |
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