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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 4104 | . . . . 5 | |
2 | dmeq 4799 | . . . . . . . 8 | |
3 | fveq1 5480 | . . . . . . . . 9 | |
4 | 3 | fveq2d 5485 | . . . . . . . 8 |
5 | 2, 4 | iuneq12d 3885 | . . . . . . 7 |
6 | 5 | uneq2d 3272 | . . . . . 6 |
7 | eqid 2164 | . . . . . 6 | |
8 | rdg.1 | . . . . . . 7 | |
9 | dm0 4813 | . . . . . . . . . 10 | |
10 | iuneq1 3874 | . . . . . . . . . 10 | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . 9 |
12 | 0iun 3918 | . . . . . . . . 9 | |
13 | 11, 12 | eqtri 2185 | . . . . . . . 8 |
14 | 13, 1 | eqeltri 2237 | . . . . . . 7 |
15 | 8, 14 | unex 4414 | . . . . . 6 |
16 | 6, 7, 15 | fvmpt 5558 | . . . . 5 |
17 | 1, 16 | ax-mp 5 | . . . 4 |
18 | 17, 15 | eqeltri 2237 | . . 3 |
19 | df-irdg 6330 | . . . 4 recs | |
20 | 19 | tfr0 6283 | . . 3 |
21 | 18, 20 | ax-mp 5 | . 2 |
22 | 13 | uneq2i 3269 | . . . 4 |
23 | 17, 22 | eqtri 2185 | . . 3 |
24 | un0 3438 | . . 3 | |
25 | 23, 24 | eqtri 2185 | . 2 |
26 | 21, 25 | eqtri 2185 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1342 wcel 2135 cvv 2722 cun 3110 c0 3405 ciun 3861 cmpt 4038 cdm 4599 cfv 5183 crdg 6329 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-iord 4339 df-on 4341 df-suc 4344 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-res 4611 df-iota 5148 df-fun 5185 df-fn 5186 df-fv 5191 df-recs 6265 df-irdg 6330 |
This theorem is referenced by: rdg0g 6348 om0 6418 |
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