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Mirrors > Home > MPE Home > Th. List > uzrdgfni | Unicode version |
Description: The recursive definition generator on upper integers is a function. See comment in om2uzrdg 12067. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 4-May-2015.) |
Ref | Expression |
---|---|
om2uz.1 | |
om2uz.2 | |
uzrdg.1 | |
uzrdg.2 | |
uzrdg.3 |
Ref | Expression |
---|---|
uzrdgfni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzrdg.3 | . . . . . . . . 9 | |
2 | 1 | eleq2i 2535 | . . . . . . . 8 |
3 | frfnom 7119 | . . . . . . . . . 10 | |
4 | uzrdg.2 | . . . . . . . . . . 11 | |
5 | 4 | fneq1i 5680 | . . . . . . . . . 10 |
6 | 3, 5 | mpbir 209 | . . . . . . . . 9 |
7 | fvelrnb 5920 | . . . . . . . . 9 | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 |
9 | 2, 8 | bitri 249 | . . . . . . 7 |
10 | om2uz.1 | . . . . . . . . . . 11 | |
11 | om2uz.2 | . . . . . . . . . . 11 | |
12 | uzrdg.1 | . . . . . . . . . . 11 | |
13 | 10, 11, 12, 4 | om2uzrdg 12067 | . . . . . . . . . 10 |
14 | 10, 11 | om2uzuzi 12060 | . . . . . . . . . . 11 |
15 | fvex 5881 | . . . . . . . . . . 11 | |
16 | opelxpi 5036 | . . . . . . . . . . 11 | |
17 | 14, 15, 16 | sylancl 662 | . . . . . . . . . 10 |
18 | 13, 17 | eqeltrd 2545 | . . . . . . . . 9 |
19 | eleq1 2529 | . . . . . . . . 9 | |
20 | 18, 19 | syl5ibcom 220 | . . . . . . . 8 |
21 | 20 | rexlimiv 2943 | . . . . . . 7 |
22 | 9, 21 | sylbi 195 | . . . . . 6 |
23 | 22 | ssriv 3507 | . . . . 5 |
24 | xpss 5114 | . . . . 5 | |
25 | 23, 24 | sstri 3512 | . . . 4 |
26 | df-rel 5011 | . . . 4 | |
27 | 25, 26 | mpbir 209 | . . 3 |
28 | fvex 5881 | . . . . . 6 | |
29 | eqeq2 2472 | . . . . . . . 8 | |
30 | 29 | imbi2d 316 | . . . . . . 7 |
31 | 30 | albidv 1713 | . . . . . 6 |
32 | 28, 31 | spcev 3201 | . . . . 5 |
33 | 1 | eleq2i 2535 | . . . . . . 7 |
34 | fvelrnb 5920 | . . . . . . . 8 | |
35 | 6, 34 | ax-mp 5 | . . . . . . 7 |
36 | 33, 35 | bitri 249 | . . . . . 6 |
37 | 13 | eqeq1d 2459 | . . . . . . . . . . . 12 |
38 | fvex 5881 | . . . . . . . . . . . . 13 | |
39 | 38, 15 | opth1 4725 | . . . . . . . . . . . 12 |
40 | 37, 39 | syl6bi 228 | . . . . . . . . . . 11 |
41 | 10, 11 | om2uzf1oi 12064 | . . . . . . . . . . . 12 |
42 | f1ocnvfv 6184 | . . . . . . . . . . . 12 | |
43 | 41, 42 | mpan 670 | . . . . . . . . . . 11 |
44 | 40, 43 | syld 44 | . . . . . . . . . 10 |
45 | fveq2 5871 | . . . . . . . . . . 11 | |
46 | 45 | fveq2d 5875 | . . . . . . . . . 10 |
47 | 44, 46 | syl6 33 | . . . . . . . . 9 |
48 | 47 | imp 429 | . . . . . . . 8 |
49 | vex 3112 | . . . . . . . . . 10 | |
50 | vex 3112 | . . . . . . . . . 10 | |
51 | 49, 50 | op2ndd 6811 | . . . . . . . . 9 |
52 | 51 | adantl 466 | . . . . . . . 8 |
53 | 48, 52 | eqtr2d 2499 | . . . . . . 7 |
54 | 53 | rexlimiva 2945 | . . . . . 6 |
55 | 36, 54 | sylbi 195 | . . . . 5 |
56 | 32, 55 | mpg 1620 | . . . 4 |
57 | 56 | ax-gen 1618 | . . 3 |
58 | dffun5 5606 | . . 3 | |
59 | 27, 57, 58 | mpbir2an 920 | . 2 |
60 | dmss 5207 | . . . . 5 | |
61 | 23, 60 | ax-mp 5 | . . . 4 |
62 | dmxpss 5443 | . . . 4 | |
63 | 61, 62 | sstri 3512 | . . 3 |
64 | 10, 11, 12, 4 | uzrdglem 12068 | . . . . . 6 |
65 | 64, 1 | syl6eleqr 2556 | . . . . 5 |
66 | 49, 28 | opeldm 5211 | . . . . 5 |
67 | 65, 66 | syl 16 | . . . 4 |
68 | 67 | ssriv 3507 | . . 3 |
69 | 63, 68 | eqssi 3519 | . 2 |
70 | df-fn 5596 | . 2 | |
71 | 59, 69, 70 | mpbir2an 920 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 <-> wb 184
/\ wa 369 A. wal 1393 = wceq 1395
E. wex 1612 e. wcel 1818 E. wrex 2808
cvv 3109
C_ wss 3475 <. cop 4035 e. cmpt 4510
X. cxp 5002 `' ccnv 5003 dom cdm 5004
ran crn 5005 |` cres 5006 Rel wrel 5009
Fun wfun 5587
Fn wfn 5588 -1-1-onto-> wf1o 5592 ` cfv 5593 (class class class)co 6296
e. cmpt2 6298 com 6700
c2nd 6799
rec crdg 7094
1 c1 9514 caddc 9516 cz 10889 cuz 11110 |
This theorem is referenced by: uzrdg0i 12070 uzrdgsuci 12071 seqfn 12119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-cnex 9569 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 ax-pre-mulgt0 9590 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-2nd 6801 df-recs 7061 df-rdg 7095 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-le 9655 df-sub 9830 df-neg 9831 df-nn 10562 df-n0 10821 df-z 10890 df-uz 11111 |
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