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Mirrors > Home > ILE Home > Th. List > rdg0 | Unicode version |
Description: The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
rdg.1 |
Ref | Expression |
---|---|
rdg0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4055 | . . . . 5 | |
2 | dmeq 4739 | . . . . . . . 8 | |
3 | fveq1 5420 | . . . . . . . . 9 | |
4 | 3 | fveq2d 5425 | . . . . . . . 8 |
5 | 2, 4 | iuneq12d 3837 | . . . . . . 7 |
6 | 5 | uneq2d 3230 | . . . . . 6 |
7 | eqid 2139 | . . . . . 6 | |
8 | rdg.1 | . . . . . . 7 | |
9 | dm0 4753 | . . . . . . . . . 10 | |
10 | iuneq1 3826 | . . . . . . . . . 10 | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . 9 |
12 | 0iun 3870 | . . . . . . . . 9 | |
13 | 11, 12 | eqtri 2160 | . . . . . . . 8 |
14 | 13, 1 | eqeltri 2212 | . . . . . . 7 |
15 | 8, 14 | unex 4362 | . . . . . 6 |
16 | 6, 7, 15 | fvmpt 5498 | . . . . 5 |
17 | 1, 16 | ax-mp 5 | . . . 4 |
18 | 17, 15 | eqeltri 2212 | . . 3 |
19 | df-irdg 6267 | . . . 4 recs | |
20 | 19 | tfr0 6220 | . . 3 |
21 | 18, 20 | ax-mp 5 | . 2 |
22 | 13 | uneq2i 3227 | . . . 4 |
23 | 17, 22 | eqtri 2160 | . . 3 |
24 | un0 3396 | . . 3 | |
25 | 23, 24 | eqtri 2160 | . 2 |
26 | 21, 25 | eqtri 2160 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 wcel 1480 cvv 2686 cun 3069 c0 3363 ciun 3813 cmpt 3989 cdm 4539 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-sep 4046 ax-nul 4054 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-ral 2421 df-rex 2422 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-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 df-recs 6202 df-irdg 6267 |
This theorem is referenced by: rdg0g 6285 om0 6354 |
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