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 10724 | . . . . 5 ♯ frec | |
2 | 1 | fveq1i 5508 | . . . 4 ♯ frec |
3 | funmpt 5246 | . . . . 5 | |
4 | funrel 5225 | . . . . . . 7 | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 |
6 | peano1 4587 | . . . . . . 7 | |
7 | reldom 6735 | . . . . . . . . . 10 | |
8 | 7 | brrelex2i 4664 | . . . . . . . . 9 |
9 | hashinfuni 10725 | . . . . . . . . . 10 | |
10 | omex 4586 | . . . . . . . . . 10 | |
11 | 9, 10 | eqeltrdi 2266 | . . . . . . . . 9 |
12 | breq2 4002 | . . . . . . . . . . . 12 | |
13 | 12 | rabbidv 2724 | . . . . . . . . . . 11 |
14 | 13 | unieqd 3816 | . . . . . . . . . 10 |
15 | eqid 2175 | . . . . . . . . . 10 | |
16 | 14, 15 | fvmptg 5584 | . . . . . . . . 9 |
17 | 8, 11, 16 | syl2anc 411 | . . . . . . . 8 |
18 | 17, 9 | eqtrd 2208 | . . . . . . 7 |
19 | 6, 18 | eleqtrrid 2265 | . . . . . 6 |
20 | relelfvdm 5539 | . . . . . 6 | |
21 | 5, 19, 20 | sylancr 414 | . . . . 5 |
22 | fvco 5578 | . . . . 5 frec frec | |
23 | 3, 21, 22 | sylancr 414 | . . . 4 frec frec |
24 | 2, 23 | eqtrid 2220 | . . 3 ♯ frec |
25 | 18 | fveq2d 5511 | . . 3 frec frec |
26 | 24, 25 | eqtrd 2208 | . 2 ♯ frec |
27 | pnfxr 7984 | . . 3 | |
28 | ordom 4600 | . . . . 5 | |
29 | ordirr 4535 | . . . . 5 | |
30 | 28, 29 | ax-mp 5 | . . . 4 |
31 | zex 9235 | . . . . . . . . . 10 | |
32 | 31 | mptex 5734 | . . . . . . . . 9 |
33 | vex 2738 | . . . . . . . . 9 | |
34 | 32, 33 | fvex 5527 | . . . . . . . 8 |
35 | 34 | ax-gen 1447 | . . . . . . 7 |
36 | 0z 9237 | . . . . . . 7 | |
37 | frecfnom 6392 | . . . . . . 7 frec | |
38 | 35, 36, 37 | mp2an 426 | . . . . . 6 frec |
39 | fndm 5307 | . . . . . 6 frec frec | |
40 | 38, 39 | ax-mp 5 | . . . . 5 frec |
41 | 40 | eleq2i 2242 | . . . 4 frec |
42 | 30, 41 | mtbir 671 | . . 3 frec |
43 | fsnunfv 5709 | . . 3 frec frec | |
44 | 10, 27, 42, 43 | mp3an 1337 | . 2 frec |
45 | 26, 44 | eqtrdi 2224 | 1 ♯ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wal 1351 wceq 1353 wcel 2146 crab 2457 cvv 2735 cun 3125 c0 3420 csn 3589 cop 3592 cuni 3805 class class class wbr 3998 cmpt 4059 word 4356 com 4583 cdm 4620 ccom 4624 wrel 4625 wfun 5202 wfn 5203 cfv 5208 (class class class)co 5865 freccfrec 6381 cdom 6729 cc0 7786 c1 7787 caddc 7789 cpnf 7963 cxr 7965 cz 9226 ♯chash 10723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1re 7880 ax-addrcl 7883 ax-rnegex 7895 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ov 5868 df-recs 6296 df-frec 6382 df-dom 6732 df-pnf 7968 df-xr 7970 df-neg 8105 df-z 9227 df-ihash 10724 |
This theorem is referenced by: filtinf 10739 |
Copyright terms: Public domain | W3C validator |