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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 6811 | . . . . 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 1132 | . . 3 β’ ((π΄ β Fin β§ πΉ Fn π΄ β§ ran πΉ = π΄) β πΉ:π΄βontoβran πΉ) |
4 | foeq3 6803 | . . . 4 β’ (ran πΉ = π΄ β (πΉ:π΄βontoβran πΉ β πΉ:π΄βontoβπ΄)) | |
5 | 4 | 3ad2ant3 1133 | . . 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 8994 | . . 3 β’ (π΄ β Fin β π΄ β π΄) | |
8 | 7 | 3ad2ant1 1131 | . 2 β’ ((π΄ β Fin β§ πΉ Fn π΄ β§ ran πΉ = π΄) β π΄ β π΄) |
9 | simp1 1134 | . 2 β’ ((π΄ β Fin β§ πΉ Fn π΄ β§ ran πΉ = π΄) β π΄ β Fin) | |
10 | fofinf1o 9341 | . 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 1369 | 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 1085 = wceq 1534 β wcel 2099 class class class wbr 5142 ran crn 5673 Fn wfn 6537 βontoβwfo 6540 β1-1-ontoβwf1o 6541 β cen 8950 Fincfn 8953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7863 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 |
This theorem is referenced by: gausslemma2dlem1 27273 |
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