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Mirrors > Home > ILE Home > Th. List > hashennn | Unicode version |
Description: The size of a set equinumerous to an element of . (Contributed by Jim Kingdon, 21-Feb-2022.) |
Ref | Expression |
---|---|
hashennn | ♯ frec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ihash 10515 | . . . . 5 ♯ frec | |
2 | 1 | fveq1i 5415 | . . . 4 ♯ frec |
3 | funmpt 5156 | . . . . 5 | |
4 | hashennnuni 10518 | . . . . . . . . 9 | |
5 | 4 | eqcomd 2143 | . . . . . . . 8 |
6 | nnfi 6759 | . . . . . . . . . . 11 | |
7 | 6 | adantr 274 | . . . . . . . . . 10 |
8 | simpr 109 | . . . . . . . . . . 11 | |
9 | 8 | ensymd 6670 | . . . . . . . . . 10 |
10 | enfii 6761 | . . . . . . . . . 10 | |
11 | 7, 9, 10 | syl2anc 408 | . . . . . . . . 9 |
12 | simpl 108 | . . . . . . . . 9 | |
13 | simpr 109 | . . . . . . . . . . 11 | |
14 | breq2 3928 | . . . . . . . . . . . . . 14 | |
15 | 14 | adantr 274 | . . . . . . . . . . . . 13 |
16 | 15 | rabbidv 2670 | . . . . . . . . . . . 12 |
17 | 16 | unieqd 3742 | . . . . . . . . . . 11 |
18 | 13, 17 | eqeq12d 2152 | . . . . . . . . . 10 |
19 | 18 | opelopabga 4180 | . . . . . . . . 9 |
20 | 11, 12, 19 | syl2anc 408 | . . . . . . . 8 |
21 | 5, 20 | mpbird 166 | . . . . . . 7 |
22 | mptv 4020 | . . . . . . 7 | |
23 | 21, 22 | eleqtrrdi 2231 | . . . . . 6 |
24 | opeldmg 4739 | . . . . . . 7 | |
25 | 11, 12, 24 | syl2anc 408 | . . . . . 6 |
26 | 23, 25 | mpd 13 | . . . . 5 |
27 | fvco 5484 | . . . . 5 frec frec | |
28 | 3, 26, 27 | sylancr 410 | . . . 4 frec frec |
29 | 2, 28 | syl5eq 2182 | . . 3 ♯ frec |
30 | 11 | elexd 2694 | . . . . . 6 |
31 | 4, 12 | eqeltrd 2214 | . . . . . 6 |
32 | 14 | rabbidv 2670 | . . . . . . . 8 |
33 | 32 | unieqd 3742 | . . . . . . 7 |
34 | eqid 2137 | . . . . . . 7 | |
35 | 33, 34 | fvmptg 5490 | . . . . . 6 |
36 | 30, 31, 35 | syl2anc 408 | . . . . 5 |
37 | 36, 4 | eqtrd 2170 | . . . 4 |
38 | 37 | fveq2d 5418 | . . 3 frec frec |
39 | 29, 38 | eqtrd 2170 | . 2 ♯ frec |
40 | ordom 4515 | . . . . . . 7 | |
41 | ordirr 4452 | . . . . . . 7 | |
42 | 40, 41 | ax-mp 5 | . . . . . 6 |
43 | eleq1 2200 | . . . . . 6 | |
44 | 42, 43 | mtbii 663 | . . . . 5 |
45 | 44 | necon2ai 2360 | . . . 4 |
46 | fvunsng 5607 | . . . 4 frec frec | |
47 | 45, 46 | mpdan 417 | . . 3 frec frec |
48 | 47 | adantr 274 | . 2 frec frec |
49 | 39, 48 | eqtrd 2170 | 1 ♯ frec |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wne 2306 crab 2418 cvv 2681 cun 3064 csn 3522 cop 3525 cuni 3731 class class class wbr 3924 copab 3983 cmpt 3984 word 4279 com 4499 cdm 4534 ccom 4538 wfun 5112 cfv 5118 (class class class)co 5767 freccfrec 6280 cen 6625 cdom 6626 cfn 6627 cc0 7613 c1 7614 caddc 7616 cpnf 7790 cz 9047 ♯chash 10514 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-ihash 10515 |
This theorem is referenced by: hashcl 10520 hashfz1 10522 hashen 10523 fihashdom 10542 hashun 10544 |
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