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Mirrors > Home > MPE Home > Th. List > wfrlem5 | Structured version Visualization version Unicode version |
Description: Lemma for well-founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
wfrlem5.1 | |
wfrlem5.2 | Se |
wfrlem5.3 |
Ref | Expression |
---|---|
wfrlem5 |
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 | wfrlem5.3 | . . . . . . . . 9 | |
16 | 15 | wfrlem3 7416 | . . . . . . . 8 |
17 | ssinss1 3841 | . . . . . . . 8 | |
18 | wfrlem5.1 | . . . . . . . . . 10 | |
19 | wess 5101 | . . . . . . . . . 10 | |
20 | 18, 19 | mpi 20 | . . . . . . . . 9 |
21 | wfrlem5.2 | . . . . . . . . . 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 | 18, 15 | wfrlem4 7418 | . . . . . 6 |
28 | 18, 15 | wfrlem4 7418 | . . . . . . . 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 | fveq2i 6194 | . . . . . . . . . 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 | wfr3g 7413 | . . . . . 6 Se | |
45 | 26, 27, 43, 44 | syl3anc 1326 | . . . . 5 |
46 | 45 | breqd 4664 | . . . 4 |
47 | 46 | biimprd 238 | . . 3 |
48 | 15 | wfrlem2 7415 | . . . . 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 Se wse 5071 wwe 5072 cdm 5114 cres 5116 wrel 5119 cpred 5679 wfun 5882 wfn 5883 cfv 5888 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
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-rmo 2920 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-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 |
This theorem is referenced by: wfrfun 7425 |
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