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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 10522 | . . . . 5 ♯ frec | |
2 | 1 | fveq1i 5422 | . . . 4 ♯ frec |
3 | funmpt 5161 | . . . . 5 | |
4 | hashennnuni 10525 | . . . . . . . . 9 | |
5 | 4 | eqcomd 2145 | . . . . . . . 8 |
6 | nnfi 6766 | . . . . . . . . . . 11 | |
7 | 6 | adantr 274 | . . . . . . . . . 10 |
8 | simpr 109 | . . . . . . . . . . 11 | |
9 | 8 | ensymd 6677 | . . . . . . . . . 10 |
10 | enfii 6768 | . . . . . . . . . 10 | |
11 | 7, 9, 10 | syl2anc 408 | . . . . . . . . 9 |
12 | simpl 108 | . . . . . . . . 9 | |
13 | simpr 109 | . . . . . . . . . . 11 | |
14 | breq2 3933 | . . . . . . . . . . . . . 14 | |
15 | 14 | adantr 274 | . . . . . . . . . . . . 13 |
16 | 15 | rabbidv 2675 | . . . . . . . . . . . 12 |
17 | 16 | unieqd 3747 | . . . . . . . . . . 11 |
18 | 13, 17 | eqeq12d 2154 | . . . . . . . . . 10 |
19 | 18 | opelopabga 4185 | . . . . . . . . 9 |
20 | 11, 12, 19 | syl2anc 408 | . . . . . . . 8 |
21 | 5, 20 | mpbird 166 | . . . . . . 7 |
22 | mptv 4025 | . . . . . . 7 | |
23 | 21, 22 | eleqtrrdi 2233 | . . . . . 6 |
24 | opeldmg 4744 | . . . . . . 7 | |
25 | 11, 12, 24 | syl2anc 408 | . . . . . 6 |
26 | 23, 25 | mpd 13 | . . . . 5 |
27 | fvco 5491 | . . . . 5 frec frec | |
28 | 3, 26, 27 | sylancr 410 | . . . 4 frec frec |
29 | 2, 28 | syl5eq 2184 | . . 3 ♯ frec |
30 | 11 | elexd 2699 | . . . . . 6 |
31 | 4, 12 | eqeltrd 2216 | . . . . . 6 |
32 | 14 | rabbidv 2675 | . . . . . . . 8 |
33 | 32 | unieqd 3747 | . . . . . . 7 |
34 | eqid 2139 | . . . . . . 7 | |
35 | 33, 34 | fvmptg 5497 | . . . . . 6 |
36 | 30, 31, 35 | syl2anc 408 | . . . . 5 |
37 | 36, 4 | eqtrd 2172 | . . . 4 |
38 | 37 | fveq2d 5425 | . . 3 frec frec |
39 | 29, 38 | eqtrd 2172 | . 2 ♯ frec |
40 | ordom 4520 | . . . . . . 7 | |
41 | ordirr 4457 | . . . . . . 7 | |
42 | 40, 41 | ax-mp 5 | . . . . . 6 |
43 | eleq1 2202 | . . . . . 6 | |
44 | 42, 43 | mtbii 663 | . . . . 5 |
45 | 44 | necon2ai 2362 | . . . 4 |
46 | fvunsng 5614 | . . . 4 frec frec | |
47 | 45, 46 | mpdan 417 | . . 3 frec frec |
48 | 47 | adantr 274 | . 2 frec frec |
49 | 39, 48 | eqtrd 2172 | 1 ♯ frec |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wne 2308 crab 2420 cvv 2686 cun 3069 csn 3527 cop 3530 cuni 3736 class class class wbr 3929 copab 3988 cmpt 3989 word 4284 com 4504 cdm 4539 ccom 4543 wfun 5117 cfv 5123 (class class class)co 5774 freccfrec 6287 cen 6632 cdom 6633 cfn 6634 cc0 7620 c1 7621 caddc 7623 cpnf 7797 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-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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 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-rab 2425 df-v 2688 df-sbc 2910 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-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-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-ihash 10522 |
This theorem is referenced by: hashcl 10527 hashfz1 10529 hashen 10530 fihashdom 10549 hashun 10551 |
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