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Theorem rneqdmfinf1o 9342
Description: Any function from a finite set onto the same set must be a bijection. (Contributed by AV, 5-Jul-2021.)
Assertion
Ref Expression
rneqdmfinf1o ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) β†’ 𝐹:𝐴–1-1-onto→𝐴)

Proof of Theorem rneqdmfinf1o
StepHypRef Expression
1 dffn4 6811 . . . . 5 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–ontoβ†’ran 𝐹)
21biimpi 215 . . . 4 (𝐹 Fn 𝐴 β†’ 𝐹:𝐴–ontoβ†’ran 𝐹)
323ad2ant2 1132 . . 3 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) β†’ 𝐹:𝐴–ontoβ†’ran 𝐹)
4 foeq3 6803 . . . 4 (ran 𝐹 = 𝐴 β†’ (𝐹:𝐴–ontoβ†’ran 𝐹 ↔ 𝐹:𝐴–onto→𝐴))
543ad2ant3 1133 . . 3 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) β†’ (𝐹:𝐴–ontoβ†’ran 𝐹 ↔ 𝐹:𝐴–onto→𝐴))
63, 5mpbid 231 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) β†’ 𝐹:𝐴–onto→𝐴)
7 enrefg 8994 . . 3 (𝐴 ∈ Fin β†’ 𝐴 β‰ˆ 𝐴)
873ad2ant1 1131 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) β†’ 𝐴 β‰ˆ 𝐴)
9 simp1 1134 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) β†’ 𝐴 ∈ Fin)
10 fofinf1o 9341 . 2 ((𝐹:𝐴–onto→𝐴 ∧ 𝐴 β‰ˆ 𝐴 ∧ 𝐴 ∈ Fin) β†’ 𝐹:𝐴–1-1-onto→𝐴)
116, 8, 9, 10syl3anc 1369 1 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) β†’ 𝐹:𝐴–1-1-onto→𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5142  ran crn 5673   Fn wfn 6537  β€“ontoβ†’wfo 6540  β€“1-1-ontoβ†’wf1o 6541   β‰ˆ 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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