Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > hashinfom | Unicode version |
Description: The value of the ♯ function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.) |
Ref | Expression |
---|---|
hashinfom | ♯ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ihash 10522 | . . . . 5 ♯ frec | |
2 | 1 | fveq1i 5422 | . . . 4 ♯ frec |
3 | funmpt 5161 | . . . . 5 | |
4 | funrel 5140 | . . . . . . 7 | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 |
6 | peano1 4508 | . . . . . . 7 | |
7 | reldom 6639 | . . . . . . . . . 10 | |
8 | 7 | brrelex2i 4583 | . . . . . . . . 9 |
9 | hashinfuni 10523 | . . . . . . . . . 10 | |
10 | omex 4507 | . . . . . . . . . 10 | |
11 | 9, 10 | eqeltrdi 2230 | . . . . . . . . 9 |
12 | breq2 3933 | . . . . . . . . . . . 12 | |
13 | 12 | rabbidv 2675 | . . . . . . . . . . 11 |
14 | 13 | unieqd 3747 | . . . . . . . . . 10 |
15 | eqid 2139 | . . . . . . . . . 10 | |
16 | 14, 15 | fvmptg 5497 | . . . . . . . . 9 |
17 | 8, 11, 16 | syl2anc 408 | . . . . . . . 8 |
18 | 17, 9 | eqtrd 2172 | . . . . . . 7 |
19 | 6, 18 | eleqtrrid 2229 | . . . . . 6 |
20 | relelfvdm 5453 | . . . . . 6 | |
21 | 5, 19, 20 | sylancr 410 | . . . . 5 |
22 | fvco 5491 | . . . . 5 frec frec | |
23 | 3, 21, 22 | sylancr 410 | . . . 4 frec frec |
24 | 2, 23 | syl5eq 2184 | . . 3 ♯ frec |
25 | 18 | fveq2d 5425 | . . 3 frec frec |
26 | 24, 25 | eqtrd 2172 | . 2 ♯ frec |
27 | pnfxr 7818 | . . 3 | |
28 | ordom 4520 | . . . . 5 | |
29 | ordirr 4457 | . . . . 5 | |
30 | 28, 29 | ax-mp 5 | . . . 4 |
31 | zex 9063 | . . . . . . . . . 10 | |
32 | 31 | mptex 5646 | . . . . . . . . 9 |
33 | vex 2689 | . . . . . . . . 9 | |
34 | 32, 33 | fvex 5441 | . . . . . . . 8 |
35 | 34 | ax-gen 1425 | . . . . . . 7 |
36 | 0z 9065 | . . . . . . 7 | |
37 | frecfnom 6298 | . . . . . . 7 frec | |
38 | 35, 36, 37 | mp2an 422 | . . . . . 6 frec |
39 | fndm 5222 | . . . . . 6 frec frec | |
40 | 38, 39 | ax-mp 5 | . . . . 5 frec |
41 | 40 | eleq2i 2206 | . . . 4 frec |
42 | 30, 41 | mtbir 660 | . . 3 frec |
43 | fsnunfv 5621 | . . 3 frec frec | |
44 | 10, 27, 42, 43 | mp3an 1315 | . 2 frec |
45 | 26, 44 | syl6eq 2188 | 1 ♯ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wal 1329 wceq 1331 wcel 1480 crab 2420 cvv 2686 cun 3069 c0 3363 csn 3527 cop 3530 cuni 3736 class class class wbr 3929 cmpt 3989 word 4284 com 4504 cdm 4539 ccom 4543 wrel 4544 wfun 5117 wfn 5118 cfv 5123 (class class class)co 5774 freccfrec 6287 cdom 6633 cc0 7620 c1 7621 caddc 7623 cpnf 7797 cxr 7799 cz 9054 ♯chash 10521 |
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 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-rnegex 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 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-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-ov 5777 df-recs 6202 df-frec 6288 df-dom 6636 df-pnf 7802 df-xr 7804 df-neg 7936 df-z 9055 df-ihash 10522 |
This theorem is referenced by: filtinf 10538 |
Copyright terms: Public domain | W3C validator |