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Theorem rneqdmfinf1o 8530
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 6372 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
21biimpi 208 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
323ad2ant2 1125 . . 3 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴onto→ran 𝐹)
4 foeq3 6364 . . . 4 (ran 𝐹 = 𝐴 → (𝐹:𝐴onto→ran 𝐹𝐹:𝐴onto𝐴))
543ad2ant3 1126 . . 3 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → (𝐹:𝐴onto→ran 𝐹𝐹:𝐴onto𝐴))
63, 5mpbid 224 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴onto𝐴)
7 enrefg 8273 . . 3 (𝐴 ∈ Fin → 𝐴𝐴)
873ad2ant1 1124 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐴𝐴)
9 simp1 1127 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐴 ∈ Fin)
10 fofinf1o 8529 . 2 ((𝐹:𝐴onto𝐴𝐴𝐴𝐴 ∈ Fin) → 𝐹:𝐴1-1-onto𝐴)
116, 8, 9, 10syl3anc 1439 1 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1071   = wceq 1601  wcel 2107   class class class wbr 4886  ran crn 5356   Fn wfn 6130  ontowfo 6133  1-1-ontowf1o 6134  cen 8238  Fincfn 8241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-om 7344  df-1o 7843  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245
This theorem is referenced by:  gausslemma2dlem1  25543
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