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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 10722 | . . . . 5 ♯ frec | |
2 | 1 | fveq1i 5508 | . . . 4 ♯ frec |
3 | funmpt 5246 | . . . . 5 | |
4 | hashennnuni 10725 | . . . . . . . . 9 | |
5 | 4 | eqcomd 2181 | . . . . . . . 8 |
6 | nnfi 6862 | . . . . . . . . . . 11 | |
7 | 6 | adantr 276 | . . . . . . . . . 10 |
8 | simpr 110 | . . . . . . . . . . 11 | |
9 | 8 | ensymd 6773 | . . . . . . . . . 10 |
10 | enfii 6864 | . . . . . . . . . 10 | |
11 | 7, 9, 10 | syl2anc 411 | . . . . . . . . 9 |
12 | simpl 109 | . . . . . . . . 9 | |
13 | simpr 110 | . . . . . . . . . . 11 | |
14 | breq2 4002 | . . . . . . . . . . . . . 14 | |
15 | 14 | adantr 276 | . . . . . . . . . . . . 13 |
16 | 15 | rabbidv 2724 | . . . . . . . . . . . 12 |
17 | 16 | unieqd 3816 | . . . . . . . . . . 11 |
18 | 13, 17 | eqeq12d 2190 | . . . . . . . . . 10 |
19 | 18 | opelopabga 4257 | . . . . . . . . 9 |
20 | 11, 12, 19 | syl2anc 411 | . . . . . . . 8 |
21 | 5, 20 | mpbird 167 | . . . . . . 7 |
22 | mptv 4095 | . . . . . . 7 | |
23 | 21, 22 | eleqtrrdi 2269 | . . . . . 6 |
24 | opeldmg 4825 | . . . . . . 7 | |
25 | 11, 12, 24 | syl2anc 411 | . . . . . 6 |
26 | 23, 25 | mpd 13 | . . . . 5 |
27 | fvco 5578 | . . . . 5 frec frec | |
28 | 3, 26, 27 | sylancr 414 | . . . 4 frec frec |
29 | 2, 28 | eqtrid 2220 | . . 3 ♯ frec |
30 | 11 | elexd 2748 | . . . . . 6 |
31 | 4, 12 | eqeltrd 2252 | . . . . . 6 |
32 | 14 | rabbidv 2724 | . . . . . . . 8 |
33 | 32 | unieqd 3816 | . . . . . . 7 |
34 | eqid 2175 | . . . . . . 7 | |
35 | 33, 34 | fvmptg 5584 | . . . . . 6 |
36 | 30, 31, 35 | syl2anc 411 | . . . . 5 |
37 | 36, 4 | eqtrd 2208 | . . . 4 |
38 | 37 | fveq2d 5511 | . . 3 frec frec |
39 | 29, 38 | eqtrd 2208 | . 2 ♯ frec |
40 | ordom 4600 | . . . . . . 7 | |
41 | ordirr 4535 | . . . . . . 7 | |
42 | 40, 41 | ax-mp 5 | . . . . . 6 |
43 | eleq1 2238 | . . . . . 6 | |
44 | 42, 43 | mtbii 674 | . . . . 5 |
45 | 44 | necon2ai 2399 | . . . 4 |
46 | fvunsng 5702 | . . . 4 frec frec | |
47 | 45, 46 | mpdan 421 | . . 3 frec frec |
48 | 47 | adantr 276 | . 2 frec frec |
49 | 39, 48 | eqtrd 2208 | 1 ♯ frec |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 104 wb 105 wceq 1353 wcel 2146 wne 2345 crab 2457 cvv 2735 cun 3125 csn 3589 cop 3592 cuni 3805 class class class wbr 3998 copab 4058 cmpt 4059 word 4356 com 4583 cdm 4620 ccom 4624 wfun 5202 cfv 5208 (class class class)co 5865 freccfrec 6381 cen 6728 cdom 6729 cfn 6730 cc0 7786 c1 7787 caddc 7789 cpnf 7963 cz 9224 ♯chash 10721 |
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-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 |
This theorem depends on definitions: df-bi 117 df-dc 835 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-rab 2462 df-v 2737 df-sbc 2961 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-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-er 6525 df-en 6731 df-dom 6732 df-fin 6733 df-ihash 10722 |
This theorem is referenced by: hashcl 10727 hashfz1 10729 hashen 10730 fihashdom 10749 hashun 10751 |
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