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Mirrors > Home > MPE Home > Th. List > hashf1rn | Structured version Visualization version GIF version |
Description: The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 4-May-2021.) |
Ref | Expression |
---|---|
hashf1rn | β’ ((π΄ β π β§ πΉ:π΄β1-1βπ΅) β (β―βπΉ) = (β―βran πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6739 | . . . . 5 β’ (πΉ:π΄β1-1βπ΅ β πΉ:π΄βΆπ΅) | |
2 | 1 | anim2i 618 | . . . 4 β’ ((π΄ β π β§ πΉ:π΄β1-1βπ΅) β (π΄ β π β§ πΉ:π΄βΆπ΅)) |
3 | 2 | ancomd 463 | . . 3 β’ ((π΄ β π β§ πΉ:π΄β1-1βπ΅) β (πΉ:π΄βΆπ΅ β§ π΄ β π)) |
4 | fex 7177 | . . 3 β’ ((πΉ:π΄βΆπ΅ β§ π΄ β π) β πΉ β V) | |
5 | 3, 4 | syl 17 | . 2 β’ ((π΄ β π β§ πΉ:π΄β1-1βπ΅) β πΉ β V) |
6 | f1o2ndf1 8055 | . . . 4 β’ (πΉ:π΄β1-1βπ΅ β (2nd βΎ πΉ):πΉβ1-1-ontoβran πΉ) | |
7 | df-2nd 7923 | . . . . . . . . 9 β’ 2nd = (π₯ β V β¦ βͺ ran {π₯}) | |
8 | 7 | funmpt2 6541 | . . . . . . . 8 β’ Fun 2nd |
9 | resfunexg 7166 | . . . . . . . 8 β’ ((Fun 2nd β§ πΉ β V) β (2nd βΎ πΉ) β V) | |
10 | 8, 5, 9 | sylancr 588 | . . . . . . 7 β’ ((π΄ β π β§ πΉ:π΄β1-1βπ΅) β (2nd βΎ πΉ) β V) |
11 | f1oeq1 6773 | . . . . . . . . . 10 β’ ((2nd βΎ πΉ) = π β ((2nd βΎ πΉ):πΉβ1-1-ontoβran πΉ β π:πΉβ1-1-ontoβran πΉ)) | |
12 | 11 | biimpd 228 | . . . . . . . . 9 β’ ((2nd βΎ πΉ) = π β ((2nd βΎ πΉ):πΉβ1-1-ontoβran πΉ β π:πΉβ1-1-ontoβran πΉ)) |
13 | 12 | eqcoms 2741 | . . . . . . . 8 β’ (π = (2nd βΎ πΉ) β ((2nd βΎ πΉ):πΉβ1-1-ontoβran πΉ β π:πΉβ1-1-ontoβran πΉ)) |
14 | 13 | adantl 483 | . . . . . . 7 β’ (((π΄ β π β§ πΉ:π΄β1-1βπ΅) β§ π = (2nd βΎ πΉ)) β ((2nd βΎ πΉ):πΉβ1-1-ontoβran πΉ β π:πΉβ1-1-ontoβran πΉ)) |
15 | 10, 14 | spcimedv 3553 | . . . . . 6 β’ ((π΄ β π β§ πΉ:π΄β1-1βπ΅) β ((2nd βΎ πΉ):πΉβ1-1-ontoβran πΉ β βπ π:πΉβ1-1-ontoβran πΉ)) |
16 | 15 | ex 414 | . . . . 5 β’ (π΄ β π β (πΉ:π΄β1-1βπ΅ β ((2nd βΎ πΉ):πΉβ1-1-ontoβran πΉ β βπ π:πΉβ1-1-ontoβran πΉ))) |
17 | 16 | com13 88 | . . . 4 β’ ((2nd βΎ πΉ):πΉβ1-1-ontoβran πΉ β (πΉ:π΄β1-1βπ΅ β (π΄ β π β βπ π:πΉβ1-1-ontoβran πΉ))) |
18 | 6, 17 | mpcom 38 | . . 3 β’ (πΉ:π΄β1-1βπ΅ β (π΄ β π β βπ π:πΉβ1-1-ontoβran πΉ)) |
19 | 18 | impcom 409 | . 2 β’ ((π΄ β π β§ πΉ:π΄β1-1βπ΅) β βπ π:πΉβ1-1-ontoβran πΉ) |
20 | hasheqf1oi 14257 | . 2 β’ (πΉ β V β (βπ π:πΉβ1-1-ontoβran πΉ β (β―βπΉ) = (β―βran πΉ))) | |
21 | 5, 19, 20 | sylc 65 | 1 β’ ((π΄ β π β§ πΉ:π΄β1-1βπ΅) β (β―βπΉ) = (β―βran πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 βwex 1782 β wcel 2107 Vcvv 3444 {csn 4587 βͺ cuni 4866 ran crn 5635 βΎ cres 5636 Fun wfun 6491 βΆwf 6493 β1-1βwf1 6494 β1-1-ontoβwf1o 6496 βcfv 6497 2nd c2nd 7921 β―chash 14236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-hash 14237 |
This theorem is referenced by: hashimarn 14346 usgrsizedg 28205 cycpmco2lem5 32028 cycpmconjslem2 32053 cyc3conja 32055 frlmdim 32363 ply1degltdim 32375 hashf1dmrn 33764 sticksstones2 40601 |
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