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Theorem rneqdmfinf1o 8784
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 6571 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
21biimpi 219 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
323ad2ant2 1131 . . 3 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴onto→ran 𝐹)
4 foeq3 6563 . . . 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 235 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴onto𝐴)
7 enrefg 8524 . . 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 8783 . 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 209  w3a 1084   = wceq 1538  wcel 2111   class class class wbr 5030  ran crn 5520   Fn wfn 6319  ontowfo 6322  1-1-ontowf1o 6323  cen 8489  Fincfn 8492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1o 8085  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496
This theorem is referenced by:  gausslemma2dlem1  25950
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