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Mirrors > Home > ILE Home > Th. List > rdgon | Unicode version |
Description: Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.) (Revised by Jim Kingdon, 13-Apr-2022.) |
Ref | Expression |
---|---|
rdgon.2 | |
rdgon.3 |
Ref | Expression |
---|---|
rdgon |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-irdg 6267 | . 2 recs | |
2 | funmpt 5161 | . . 3 | |
3 | 2 | a1i 9 | . 2 |
4 | ordon 4402 | . . 3 | |
5 | 4 | a1i 9 | . 2 |
6 | vex 2689 | . . . 4 | |
7 | rdgon.2 | . . . . . . 7 | |
8 | 7 | adantr 274 | . . . . . 6 |
9 | 8 | 3ad2ant1 1002 | . . . . 5 |
10 | 6 | dmex 4805 | . . . . . 6 |
11 | fveq2 5421 | . . . . . . . . . 10 | |
12 | 11 | eleq1d 2208 | . . . . . . . . 9 |
13 | rdgon.3 | . . . . . . . . . . . 12 | |
14 | 13 | adantr 274 | . . . . . . . . . . 11 |
15 | 14 | 3ad2ant1 1002 | . . . . . . . . . 10 |
16 | 15 | adantr 274 | . . . . . . . . 9 |
17 | simpl3 986 | . . . . . . . . . 10 | |
18 | simpr 109 | . . . . . . . . . . 11 | |
19 | fdm 5278 | . . . . . . . . . . . . 13 | |
20 | 19 | eleq2d 2209 | . . . . . . . . . . . 12 |
21 | 17, 20 | syl 14 | . . . . . . . . . . 11 |
22 | 18, 21 | mpbid 146 | . . . . . . . . . 10 |
23 | 17, 22 | ffvelrnd 5556 | . . . . . . . . 9 |
24 | 12, 16, 23 | rspcdva 2794 | . . . . . . . 8 |
25 | 24 | ralrimiva 2505 | . . . . . . 7 |
26 | fveq2 5421 | . . . . . . . . . 10 | |
27 | 26 | fveq2d 5425 | . . . . . . . . 9 |
28 | 27 | eleq1d 2208 | . . . . . . . 8 |
29 | 28 | cbvralv 2654 | . . . . . . 7 |
30 | 25, 29 | sylibr 133 | . . . . . 6 |
31 | iunon 6181 | . . . . . 6 | |
32 | 10, 30, 31 | sylancr 410 | . . . . 5 |
33 | onun2 4406 | . . . . 5 | |
34 | 9, 32, 33 | syl2anc 408 | . . . 4 |
35 | dmeq 4739 | . . . . . . 7 | |
36 | fveq1 5420 | . . . . . . . 8 | |
37 | 36 | fveq2d 5425 | . . . . . . 7 |
38 | 35, 37 | iuneq12d 3837 | . . . . . 6 |
39 | 38 | uneq2d 3230 | . . . . 5 |
40 | eqid 2139 | . . . . 5 | |
41 | 39, 40 | fvmptg 5497 | . . . 4 |
42 | 6, 34, 41 | sylancr 410 | . . 3 |
43 | 42, 34 | eqeltrd 2216 | . 2 |
44 | unon 4427 | . . . . . 6 | |
45 | 44 | eleq2i 2206 | . . . . 5 |
46 | 45 | biimpi 119 | . . . 4 |
47 | 46 | adantl 275 | . . 3 |
48 | suceloni 4417 | . . 3 | |
49 | 47, 48 | syl 14 | . 2 |
50 | 44 | eleq2i 2206 | . . . 4 |
51 | 50 | biimpri 132 | . . 3 |
52 | 51 | adantl 275 | . 2 |
53 | 1, 3, 5, 43, 49, 52 | tfrcl 6261 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2416 cvv 2686 cun 3069 cuni 3736 ciun 3813 cmpt 3989 word 4284 con0 4285 csuc 4287 cdm 4539 wfun 5117 wf 5119 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: oacl 6356 omcl 6357 oeicl 6358 |
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