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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 10490 | . . . . 5 ♯ frec | |
2 | 1 | fveq1i 5390 | . . . 4 ♯ frec |
3 | funmpt 5131 | . . . . 5 | |
4 | funrel 5110 | . . . . . . 7 | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 |
6 | peano1 4478 | . . . . . . 7 | |
7 | reldom 6607 | . . . . . . . . . 10 | |
8 | 7 | brrelex2i 4553 | . . . . . . . . 9 |
9 | hashinfuni 10491 | . . . . . . . . . 10 | |
10 | omex 4477 | . . . . . . . . . 10 | |
11 | 9, 10 | syl6eqel 2208 | . . . . . . . . 9 |
12 | breq2 3903 | . . . . . . . . . . . 12 | |
13 | 12 | rabbidv 2649 | . . . . . . . . . . 11 |
14 | 13 | unieqd 3717 | . . . . . . . . . 10 |
15 | eqid 2117 | . . . . . . . . . 10 | |
16 | 14, 15 | fvmptg 5465 | . . . . . . . . 9 |
17 | 8, 11, 16 | syl2anc 408 | . . . . . . . 8 |
18 | 17, 9 | eqtrd 2150 | . . . . . . 7 |
19 | 6, 18 | eleqtrrid 2207 | . . . . . 6 |
20 | relelfvdm 5421 | . . . . . 6 | |
21 | 5, 19, 20 | sylancr 410 | . . . . 5 |
22 | fvco 5459 | . . . . 5 frec frec | |
23 | 3, 21, 22 | sylancr 410 | . . . 4 frec frec |
24 | 2, 23 | syl5eq 2162 | . . 3 ♯ frec |
25 | 18 | fveq2d 5393 | . . 3 frec frec |
26 | 24, 25 | eqtrd 2150 | . 2 ♯ frec |
27 | pnfxr 7786 | . . 3 | |
28 | ordom 4490 | . . . . 5 | |
29 | ordirr 4427 | . . . . 5 | |
30 | 28, 29 | ax-mp 5 | . . . 4 |
31 | zex 9031 | . . . . . . . . . 10 | |
32 | 31 | mptex 5614 | . . . . . . . . 9 |
33 | vex 2663 | . . . . . . . . 9 | |
34 | 32, 33 | fvex 5409 | . . . . . . . 8 |
35 | 34 | ax-gen 1410 | . . . . . . 7 |
36 | 0z 9033 | . . . . . . 7 | |
37 | frecfnom 6266 | . . . . . . 7 frec | |
38 | 35, 36, 37 | mp2an 422 | . . . . . 6 frec |
39 | fndm 5192 | . . . . . 6 frec frec | |
40 | 38, 39 | ax-mp 5 | . . . . 5 frec |
41 | 40 | eleq2i 2184 | . . . 4 frec |
42 | 30, 41 | mtbir 645 | . . 3 frec |
43 | fsnunfv 5589 | . . 3 frec frec | |
44 | 10, 27, 42, 43 | mp3an 1300 | . 2 frec |
45 | 26, 44 | syl6eq 2166 | 1 ♯ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wal 1314 wceq 1316 wcel 1465 crab 2397 cvv 2660 cun 3039 c0 3333 csn 3497 cop 3500 cuni 3706 class class class wbr 3899 cmpt 3959 word 4254 com 4474 cdm 4509 ccom 4513 wrel 4514 wfun 5087 wfn 5088 cfv 5093 (class class class)co 5742 freccfrec 6255 cdom 6601 cc0 7588 c1 7589 caddc 7591 cpnf 7765 cxr 7767 cz 9022 ♯chash 10489 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 ax-rnegex 7697 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-recs 6170 df-frec 6256 df-dom 6604 df-pnf 7770 df-xr 7772 df-neg 7904 df-z 9023 df-ihash 10490 |
This theorem is referenced by: filtinf 10506 |
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