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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 10678 | . . . . 5 ♯ frec | |
2 | 1 | fveq1i 5481 | . . . 4 ♯ frec |
3 | funmpt 5220 | . . . . 5 | |
4 | hashennnuni 10681 | . . . . . . . . 9 | |
5 | 4 | eqcomd 2170 | . . . . . . . 8 |
6 | nnfi 6829 | . . . . . . . . . . 11 | |
7 | 6 | adantr 274 | . . . . . . . . . 10 |
8 | simpr 109 | . . . . . . . . . . 11 | |
9 | 8 | ensymd 6740 | . . . . . . . . . 10 |
10 | enfii 6831 | . . . . . . . . . 10 | |
11 | 7, 9, 10 | syl2anc 409 | . . . . . . . . 9 |
12 | simpl 108 | . . . . . . . . 9 | |
13 | simpr 109 | . . . . . . . . . . 11 | |
14 | breq2 3980 | . . . . . . . . . . . . . 14 | |
15 | 14 | adantr 274 | . . . . . . . . . . . . 13 |
16 | 15 | rabbidv 2710 | . . . . . . . . . . . 12 |
17 | 16 | unieqd 3794 | . . . . . . . . . . 11 |
18 | 13, 17 | eqeq12d 2179 | . . . . . . . . . 10 |
19 | 18 | opelopabga 4235 | . . . . . . . . 9 |
20 | 11, 12, 19 | syl2anc 409 | . . . . . . . 8 |
21 | 5, 20 | mpbird 166 | . . . . . . 7 |
22 | mptv 4073 | . . . . . . 7 | |
23 | 21, 22 | eleqtrrdi 2258 | . . . . . 6 |
24 | opeldmg 4803 | . . . . . . 7 | |
25 | 11, 12, 24 | syl2anc 409 | . . . . . 6 |
26 | 23, 25 | mpd 13 | . . . . 5 |
27 | fvco 5550 | . . . . 5 frec frec | |
28 | 3, 26, 27 | sylancr 411 | . . . 4 frec frec |
29 | 2, 28 | syl5eq 2209 | . . 3 ♯ frec |
30 | 11 | elexd 2734 | . . . . . 6 |
31 | 4, 12 | eqeltrd 2241 | . . . . . 6 |
32 | 14 | rabbidv 2710 | . . . . . . . 8 |
33 | 32 | unieqd 3794 | . . . . . . 7 |
34 | eqid 2164 | . . . . . . 7 | |
35 | 33, 34 | fvmptg 5556 | . . . . . 6 |
36 | 30, 31, 35 | syl2anc 409 | . . . . 5 |
37 | 36, 4 | eqtrd 2197 | . . . 4 |
38 | 37 | fveq2d 5484 | . . 3 frec frec |
39 | 29, 38 | eqtrd 2197 | . 2 ♯ frec |
40 | ordom 4578 | . . . . . . 7 | |
41 | ordirr 4513 | . . . . . . 7 | |
42 | 40, 41 | ax-mp 5 | . . . . . 6 |
43 | eleq1 2227 | . . . . . 6 | |
44 | 42, 43 | mtbii 664 | . . . . 5 |
45 | 44 | necon2ai 2388 | . . . 4 |
46 | fvunsng 5673 | . . . 4 frec frec | |
47 | 45, 46 | mpdan 418 | . . 3 frec frec |
48 | 47 | adantr 274 | . 2 frec frec |
49 | 39, 48 | eqtrd 2197 | 1 ♯ frec |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wne 2334 crab 2446 cvv 2721 cun 3109 csn 3570 cop 3573 cuni 3783 class class class wbr 3976 copab 4036 cmpt 4037 word 4334 com 4561 cdm 4598 ccom 4602 wfun 5176 cfv 5182 (class class class)co 5836 freccfrec 6349 cen 6695 cdom 6696 cfn 6697 cc0 7744 c1 7745 caddc 7747 cpnf 7921 cz 9182 ♯chash 10677 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-ihash 10678 |
This theorem is referenced by: hashcl 10683 hashfz1 10685 hashen 10686 fihashdom 10705 hashun 10707 |
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