MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rneqdmfinf1o Structured version   Visualization version   GIF version

Theorem rneqdmfinf1o 9366
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) → 𝐹:𝐴1-1-onto𝐴)

Proof of Theorem rneqdmfinf1o
StepHypRef Expression
1 dffn4 6811 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
21biimpi 215 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
323ad2ant2 1131 . . 3 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴onto→ran 𝐹)
4 foeq3 6803 . . . 4 (ran 𝐹 = 𝐴 → (𝐹:𝐴onto→ran 𝐹𝐹:𝐴onto𝐴))
543ad2ant3 1132 . . 3 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → (𝐹:𝐴onto→ran 𝐹𝐹:𝐴onto𝐴))
63, 5mpbid 231 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴onto𝐴)
7 enrefg 9005 . . 3 (𝐴 ∈ Fin → 𝐴𝐴)
873ad2ant1 1130 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐴𝐴)
9 simp1 1133 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐴 ∈ Fin)
10 fofinf1o 9365 . 2 ((𝐹:𝐴onto𝐴𝐴𝐴𝐴 ∈ Fin) → 𝐹:𝐴1-1-onto𝐴)
116, 8, 9, 10syl3anc 1368 1 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1534  wcel 2099   class class class wbr 5144  ran crn 5674   Fn wfn 6539  ontowfo 6542  1-1-ontowf1o 6543  cen 8961  Fincfn 8964
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 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  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 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7867  df-1o 8486  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968
This theorem is referenced by:  gausslemma2dlem1  27390
  Copyright terms: Public domain W3C validator