Theorem rneqdmfinf1o 9373

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:𝐴–1-1-onto→𝐴)

Proof of Theorem rneqdmfinf1o

Step	Hyp	Ref	Expression
1		dffn4 6826	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
2	1	biimpi 216	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹:𝐴–onto→ran 𝐹)
3	2	3ad2ant2 1135	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴–onto→ran 𝐹)
4		foeq3 6818	. . . 4 ⊢ (ran 𝐹 = 𝐴 → (𝐹:𝐴–onto→ran 𝐹 ↔ 𝐹:𝐴–onto→𝐴))
5	4	3ad2ant3 1136	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → (𝐹:𝐴–onto→ran 𝐹 ↔ 𝐹:𝐴–onto→𝐴))
6	3, 5	mpbid 232	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴–onto→𝐴)
7		enrefg 9024	. . 3 ⊢ (𝐴 ∈ Fin → 𝐴 ≈ 𝐴)
8	7	3ad2ant1 1134	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐴 ≈ 𝐴)
9		simp1 1137	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐴 ∈ Fin)
10		fofinf1o 9372	. 2 ⊢ ((𝐹:𝐴–onto→𝐴 ∧ 𝐴 ≈ 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹:𝐴–1-1-onto→𝐴)
11	6, 8, 9, 10	syl3anc 1373	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   class class class wbr 5143  ran crn 5686   Fn wfn 6556  –onto→wfo 6559  –1-1-onto→wf1o 6560   ≈ cen 8982  Fincfn 8985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989

This theorem is referenced by:  gausslemma2dlem1  27410