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Mirrors > Home > ILE Home > Th. List > frecsuc | Unicode version |
Description: The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.) |
Ref | Expression |
---|---|
frecsuc | frec frec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 4739 | . . . . . . . . 9 | |
2 | 1 | eqeq1d 2148 | . . . . . . . 8 |
3 | fveq1 5420 | . . . . . . . . . 10 | |
4 | 3 | fveq2d 5425 | . . . . . . . . 9 |
5 | 4 | eleq2d 2209 | . . . . . . . 8 |
6 | 2, 5 | anbi12d 464 | . . . . . . 7 |
7 | 6 | rexbidv 2438 | . . . . . 6 |
8 | 1 | eqeq1d 2148 | . . . . . . 7 |
9 | 8 | anbi1d 460 | . . . . . 6 |
10 | 7, 9 | orbi12d 782 | . . . . 5 |
11 | 10 | abbidv 2257 | . . . 4 |
12 | 11 | cbvmptv 4024 | . . 3 |
13 | eleq1 2202 | . . . . . . . 8 | |
14 | 13 | anbi2d 459 | . . . . . . 7 |
15 | 14 | rexbidv 2438 | . . . . . 6 |
16 | eleq1 2202 | . . . . . . 7 | |
17 | 16 | anbi2d 459 | . . . . . 6 |
18 | 15, 17 | orbi12d 782 | . . . . 5 |
19 | 18 | cbvabv 2264 | . . . 4 |
20 | 19 | mpteq2i 4015 | . . 3 |
21 | suceq 4324 | . . . . . . . . 9 | |
22 | 21 | eqeq2d 2151 | . . . . . . . 8 |
23 | fveq2 5421 | . . . . . . . . . 10 | |
24 | 23 | fveq2d 5425 | . . . . . . . . 9 |
25 | 24 | eleq2d 2209 | . . . . . . . 8 |
26 | 22, 25 | anbi12d 464 | . . . . . . 7 |
27 | 26 | cbvrexv 2655 | . . . . . 6 |
28 | 27 | orbi1i 752 | . . . . 5 |
29 | 28 | abbii 2255 | . . . 4 |
30 | 29 | mpteq2i 4015 | . . 3 |
31 | 12, 20, 30 | 3eqtri 2164 | . 2 |
32 | 31 | frecsuclem 6303 | 1 frec frec |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 w3a 962 wceq 1331 wcel 1480 cab 2125 wral 2416 wrex 2417 cvv 2686 c0 3363 cmpt 3989 csuc 4287 com 4504 cdm 4539 cfv 5123 freccfrec 6287 |
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-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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-int 3772 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-ilim 4291 df-suc 4293 df-iom 4505 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-frec 6288 |
This theorem is referenced by: frecrdg 6305 frec2uzsucd 10174 frec2uzrdg 10182 frecuzrdgsuc 10187 frecuzrdgg 10189 frecuzrdgsuctlem 10196 seq3val 10231 seqvalcd 10232 |
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