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Mirrors > Home > MPE Home > Th. List > rneqdmfinf1o | Structured version Visualization version GIF version |
Description: Any function from a finite set onto the same set must be a bijection. (Contributed by AV, 5-Jul-2021.) |
Ref | Expression |
---|---|
rneqdmfinf1o | ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴–1-1-onto→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn4 6372 | . . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
2 | 1 | biimpi 208 | . . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹:𝐴–onto→ran 𝐹) |
3 | 2 | 3ad2ant2 1125 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴–onto→ran 𝐹) |
4 | foeq3 6364 | . . . 4 ⊢ (ran 𝐹 = 𝐴 → (𝐹:𝐴–onto→ran 𝐹 ↔ 𝐹:𝐴–onto→𝐴)) | |
5 | 4 | 3ad2ant3 1126 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → (𝐹:𝐴–onto→ran 𝐹 ↔ 𝐹:𝐴–onto→𝐴)) |
6 | 3, 5 | mpbid 224 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴–onto→𝐴) |
7 | enrefg 8273 | . . 3 ⊢ (𝐴 ∈ Fin → 𝐴 ≈ 𝐴) | |
8 | 7 | 3ad2ant1 1124 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐴 ≈ 𝐴) |
9 | simp1 1127 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐴 ∈ Fin) | |
10 | fofinf1o 8529 | . 2 ⊢ ((𝐹:𝐴–onto→𝐴 ∧ 𝐴 ≈ 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹:𝐴–1-1-onto→𝐴) | |
11 | 6, 8, 9, 10 | syl3anc 1439 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴–1-1-onto→𝐴) |
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 –onto→wfo 6133 –1-1-onto→wf1o 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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