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Theorem rneqdmfinf1o 8407
 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 6282 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
21biimpi 206 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
323ad2ant2 1129 . . 3 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴onto→ran 𝐹)
4 foeq3 6274 . . . 4 (ran 𝐹 = 𝐴 → (𝐹:𝐴onto→ran 𝐹𝐹:𝐴onto𝐴))
543ad2ant3 1130 . . 3 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → (𝐹:𝐴onto→ran 𝐹𝐹:𝐴onto𝐴))
63, 5mpbid 222 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴onto𝐴)
7 enrefg 8153 . . 3 (𝐴 ∈ Fin → 𝐴𝐴)
873ad2ant1 1128 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐴𝐴)
9 simp1 1131 . 2 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐴 ∈ Fin)
10 fofinf1o 8406 . 2 ((𝐹:𝐴onto𝐴𝐴𝐴𝐴 ∈ Fin) → 𝐹:𝐴1-1-onto𝐴)
116, 8, 9, 10syl3anc 1477 1 ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴1-1-onto𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   class class class wbr 4804  ran crn 5267   Fn wfn 6044  –onto→wfo 6047  –1-1-onto→wf1o 6048   ≈ cen 8118  Fincfn 8121 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-1o 7729  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125 This theorem is referenced by:  gausslemma2dlem1  25290
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