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Mirrors > Home > ILE Home > Th. List > fihasheqf1oi | Unicode version |
Description: The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.) |
Ref | Expression |
---|---|
fihasheqf1oi | ♯ ♯ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1odm 5418 | . . . . . 6 | |
2 | 1 | eleq1d 2226 | . . . . 5 |
3 | 2 | biimparc 297 | . . . 4 |
4 | f1ofun 5416 | . . . . . 6 | |
5 | 4 | adantl 275 | . . . . 5 |
6 | fundmfibi 6883 | . . . . 5 | |
7 | 5, 6 | syl 14 | . . . 4 |
8 | 3, 7 | mpbird 166 | . . 3 |
9 | simpr 109 | . . 3 | |
10 | f1oeq1 5403 | . . . 4 | |
11 | 10 | spcegv 2800 | . . 3 |
12 | 8, 9, 11 | sylc 62 | . 2 |
13 | f1ofi 6887 | . . 3 | |
14 | hasheqf1o 10659 | . . 3 ♯ ♯ | |
15 | 13, 14 | syldan 280 | . 2 ♯ ♯ |
16 | 12, 15 | mpbird 166 | 1 ♯ ♯ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 cdm 4586 wfun 5164 wf1o 5169 cfv 5170 cfn 6685 ♯chash 10649 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-recs 6252 df-frec 6338 df-1o 6363 df-er 6480 df-en 6686 df-dom 6687 df-fin 6688 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 df-n0 9091 df-z 9168 df-uz 9440 df-ihash 10650 |
This theorem is referenced by: fihashf1rn 10663 fihasheqf1od 10664 fsum3 11284 |
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