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Mirrors > Home > MPE Home > Th. List > rneqdmfinf1o | Structured version Visualization version GIF version |
Description: Any function from a finite set onto the same set must be a bijection. (Contributed by AV, 5-Jul-2021.) |
Ref | Expression |
---|---|
rneqdmfinf1o | β’ ((π΄ β Fin β§ πΉ Fn π΄ β§ ran πΉ = π΄) β πΉ:π΄β1-1-ontoβπ΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn4 6810 | . . . . 5 β’ (πΉ Fn π΄ β πΉ:π΄βontoβran πΉ) | |
2 | 1 | biimpi 215 | . . . 4 β’ (πΉ Fn π΄ β πΉ:π΄βontoβran πΉ) |
3 | 2 | 3ad2ant2 1131 | . . 3 β’ ((π΄ β Fin β§ πΉ Fn π΄ β§ ran πΉ = π΄) β πΉ:π΄βontoβran πΉ) |
4 | foeq3 6802 | . . . 4 β’ (ran πΉ = π΄ β (πΉ:π΄βontoβran πΉ β πΉ:π΄βontoβπ΄)) | |
5 | 4 | 3ad2ant3 1132 | . . 3 β’ ((π΄ β Fin β§ πΉ Fn π΄ β§ ran πΉ = π΄) β (πΉ:π΄βontoβran πΉ β πΉ:π΄βontoβπ΄)) |
6 | 3, 5 | mpbid 231 | . 2 β’ ((π΄ β Fin β§ πΉ Fn π΄ β§ ran πΉ = π΄) β πΉ:π΄βontoβπ΄) |
7 | enrefg 8998 | . . 3 β’ (π΄ β Fin β π΄ β π΄) | |
8 | 7 | 3ad2ant1 1130 | . 2 β’ ((π΄ β Fin β§ πΉ Fn π΄ β§ ran πΉ = π΄) β π΄ β π΄) |
9 | simp1 1133 | . 2 β’ ((π΄ β Fin β§ πΉ Fn π΄ β§ ran πΉ = π΄) β π΄ β Fin) | |
10 | fofinf1o 9346 | . 2 β’ ((πΉ:π΄βontoβπ΄ β§ π΄ β π΄ β§ π΄ β Fin) β πΉ:π΄β1-1-ontoβπ΄) | |
11 | 6, 8, 9, 10 | syl3anc 1368 | 1 β’ ((π΄ β Fin β§ πΉ Fn π΄ β§ ran πΉ = π΄) β πΉ:π΄β1-1-ontoβπ΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5144 ran crn 5674 Fn wfn 6538 βontoβwfo 6541 β1-1-ontoβwf1o 6542 β cen 8954 Fincfn 8957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7866 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 |
This theorem is referenced by: gausslemma2dlem1 27312 |
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