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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 4050 | . . . . 5 | |
2 | dmeq 4734 | . . . . . . . 8 | |
3 | fveq1 5413 | . . . . . . . . 9 | |
4 | 3 | fveq2d 5418 | . . . . . . . 8 |
5 | 2, 4 | iuneq12d 3832 | . . . . . . 7 |
6 | 5 | uneq2d 3225 | . . . . . 6 |
7 | eqid 2137 | . . . . . 6 | |
8 | rdg.1 | . . . . . . 7 | |
9 | dm0 4748 | . . . . . . . . . 10 | |
10 | iuneq1 3821 | . . . . . . . . . 10 | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . 9 |
12 | 0iun 3865 | . . . . . . . . 9 | |
13 | 11, 12 | eqtri 2158 | . . . . . . . 8 |
14 | 13, 1 | eqeltri 2210 | . . . . . . 7 |
15 | 8, 14 | unex 4357 | . . . . . 6 |
16 | 6, 7, 15 | fvmpt 5491 | . . . . 5 |
17 | 1, 16 | ax-mp 5 | . . . 4 |
18 | 17, 15 | eqeltri 2210 | . . 3 |
19 | df-irdg 6260 | . . . 4 recs | |
20 | 19 | tfr0 6213 | . . 3 |
21 | 18, 20 | ax-mp 5 | . 2 |
22 | 13 | uneq2i 3222 | . . . 4 |
23 | 17, 22 | eqtri 2158 | . . 3 |
24 | un0 3391 | . . 3 | |
25 | 23, 24 | eqtri 2158 | . 2 |
26 | 21, 25 | eqtri 2158 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 wcel 1480 cvv 2681 cun 3064 c0 3358 ciun 3808 cmpt 3984 cdm 4534 cfv 5118 crdg 6259 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-res 4546 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 df-recs 6195 df-irdg 6260 |
This theorem is referenced by: rdg0g 6278 om0 6347 |
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