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Mirrors > Home > MPE Home > Th. List > Mathboxes > frrlem5 | Structured version Visualization version Unicode version |
Description: Lemma for founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Paul Chapman, 21-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
frrlem5.1 | |
frrlem5.2 | Se |
frrlem5.3 |
Ref | Expression |
---|---|
frrlem5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3203 | . . . . . 6 | |
2 | vex 3203 | . . . . . 6 | |
3 | 1, 2 | breldm 5329 | . . . . 5 |
4 | vex 3203 | . . . . . 6 | |
5 | 1, 4 | breldm 5329 | . . . . 5 |
6 | 3, 5 | anim12i 590 | . . . 4 |
7 | elin 3796 | . . . 4 | |
8 | 6, 7 | sylibr 224 | . . 3 |
9 | anandir 872 | . . . 4 | |
10 | 2 | brres 5402 | . . . . 5 |
11 | 4 | brres 5402 | . . . . 5 |
12 | 10, 11 | anbi12i 733 | . . . 4 |
13 | 9, 12 | sylbb2 228 | . . 3 |
14 | 8, 13 | mpdan 702 | . 2 |
15 | frrlem5.3 | . . . . . . . . 9 | |
16 | 15 | frrlem3 31782 | . . . . . . . 8 |
17 | ssinss1 3841 | . . . . . . . 8 | |
18 | frrlem5.1 | . . . . . . . . . 10 | |
19 | frss 5081 | . . . . . . . . . 10 | |
20 | 18, 19 | mpi 20 | . . . . . . . . 9 |
21 | frrlem5.2 | . . . . . . . . . 10 Se | |
22 | sess2 5083 | . . . . . . . . . 10 Se Se | |
23 | 21, 22 | mpi 20 | . . . . . . . . 9 Se |
24 | 20, 23 | jca 554 | . . . . . . . 8 Se |
25 | 16, 17, 24 | 3syl 18 | . . . . . . 7 Se |
26 | 25 | adantr 481 | . . . . . 6 Se |
27 | 15 | frrlem4 31783 | . . . . . 6 |
28 | 15 | frrlem4 31783 | . . . . . . . 8 |
29 | 28 | ancoms 469 | . . . . . . 7 |
30 | incom 3805 | . . . . . . . . . . 11 | |
31 | 30 | reseq2i 5393 | . . . . . . . . . 10 |
32 | 31 | fneq1i 5985 | . . . . . . . . 9 |
33 | 30 | fneq2i 5986 | . . . . . . . . 9 |
34 | 32, 33 | bitri 264 | . . . . . . . 8 |
35 | 31 | fveq1i 6192 | . . . . . . . . . 10 |
36 | predeq2 5683 | . . . . . . . . . . . . 13 | |
37 | 30, 36 | ax-mp 5 | . . . . . . . . . . . 12 |
38 | 31, 37 | reseq12i 5394 | . . . . . . . . . . 11 |
39 | 38 | oveq2i 6661 | . . . . . . . . . 10 |
40 | 35, 39 | eqeq12i 2636 | . . . . . . . . 9 |
41 | 30, 40 | raleqbii 2990 | . . . . . . . 8 |
42 | 34, 41 | anbi12i 733 | . . . . . . 7 |
43 | 29, 42 | sylibr 224 | . . . . . 6 |
44 | frr3g 31779 | . . . . . 6 Se | |
45 | 26, 27, 43, 44 | syl3anc 1326 | . . . . 5 |
46 | 45 | breqd 4664 | . . . 4 |
47 | 46 | biimprd 238 | . . 3 |
48 | 15 | frrlem2 31781 | . . . . 5 |
49 | funres 5929 | . . . . 5 | |
50 | dffun2 5898 | . . . . . 6 | |
51 | 50 | simprbi 480 | . . . . 5 |
52 | 2sp 2056 | . . . . . 6 | |
53 | 52 | sps 2055 | . . . . 5 |
54 | 48, 49, 51, 53 | 4syl 19 | . . . 4 |
55 | 54 | adantr 481 | . . 3 |
56 | 47, 55 | sylan2d 499 | . 2 |
57 | 14, 56 | syl5 34 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wal 1481 wceq 1483 wex 1704 wcel 1990 cab 2608 wral 2912 cin 3573 wss 3574 class class class wbr 4653 wfr 5070 Se wse 5071 cdm 5114 cres 5116 wrel 5119 cpred 5679 wfun 5882 wfn 5883 cfv 5888 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-trpred 31718 |
This theorem is referenced by: frrlem5c 31786 |
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