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Theorem f1finf1o 9267
Description: Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.) Avoid ax-pow 5362. (Revised by BTernaryTau, 4-Jan-2025.)
Assertion
Ref Expression
f1finf1o ((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) β†’ (𝐹:𝐴–1-1→𝐡 ↔ 𝐹:𝐴–1-1-onto→𝐡))
Proof of Theorem f1finf1o
StepHypRef Expression
1 simpr 485 . . . 4 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴–1-1→𝐡)
2 f1f 6784 . . . . . . 7 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴⟢𝐡)
32adantl 482 . . . . . 6 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴⟢𝐡)
43ffnd 6715 . . . . 5 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹 Fn 𝐴)
5 simpll 765 . . . . . 6 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐴 β‰ˆ 𝐡)
63frnd 6722 . . . . . . . . 9 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ ran 𝐹 βŠ† 𝐡)
7 df-pss 3966 . . . . . . . . . 10 (ran 𝐹 ⊊ 𝐡 ↔ (ran 𝐹 βŠ† 𝐡 ∧ ran 𝐹 β‰  𝐡))
87baib 536 . . . . . . . . 9 (ran 𝐹 βŠ† 𝐡 β†’ (ran 𝐹 ⊊ 𝐡 ↔ ran 𝐹 β‰  𝐡))
96, 8syl 17 . . . . . . . 8 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 ⊊ 𝐡 ↔ ran 𝐹 β‰  𝐡))
10 php3 9208 . . . . . . . . . . . 12 ((𝐡 ∈ Fin ∧ ran 𝐹 ⊊ 𝐡) β†’ ran 𝐹 β‰Ί 𝐡)
1110ex 413 . . . . . . . . . . 11 (𝐡 ∈ Fin β†’ (ran 𝐹 ⊊ 𝐡 β†’ ran 𝐹 β‰Ί 𝐡))
1211ad2antlr 725 . . . . . . . . . 10 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 ⊊ 𝐡 β†’ ran 𝐹 β‰Ί 𝐡))
13 enfii 9185 . . . . . . . . . . . 12 ((𝐡 ∈ Fin ∧ 𝐴 β‰ˆ 𝐡) β†’ 𝐴 ∈ Fin)
1413ancoms 459 . . . . . . . . . . 11 ((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) β†’ 𝐴 ∈ Fin)
15 f1f1orn 6841 . . . . . . . . . . . 12 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
16 f1oenfi 9178 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹) β†’ 𝐴 β‰ˆ ran 𝐹)
1714, 15, 16syl2an 596 . . . . . . . . . . 11 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐴 β‰ˆ ran 𝐹)
18 endom 8971 . . . . . . . . . . . . 13 (𝐴 β‰ˆ ran 𝐹 β†’ 𝐴 β‰Ό ran 𝐹)
19 domsdomtrfi 9201 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝐴 β‰Ό ran 𝐹 ∧ ran 𝐹 β‰Ί 𝐡) β†’ 𝐴 β‰Ί 𝐡)
2018, 19syl3an2 1164 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝐴 β‰ˆ ran 𝐹 ∧ ran 𝐹 β‰Ί 𝐡) β†’ 𝐴 β‰Ί 𝐡)
21203expia 1121 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ 𝐴 β‰ˆ ran 𝐹) β†’ (ran 𝐹 β‰Ί 𝐡 β†’ 𝐴 β‰Ί 𝐡))
2214, 17, 21syl2an2r 683 . . . . . . . . . 10 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 β‰Ί 𝐡 β†’ 𝐴 β‰Ί 𝐡))
2312, 22syld 47 . . . . . . . . 9 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 ⊊ 𝐡 β†’ 𝐴 β‰Ί 𝐡))
24 sdomnen 8973 . . . . . . . . 9 (𝐴 β‰Ί 𝐡 β†’ Β¬ 𝐴 β‰ˆ 𝐡)
2523, 24syl6 35 . . . . . . . 8 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 ⊊ 𝐡 β†’ Β¬ 𝐴 β‰ˆ 𝐡))
269, 25sylbird 259 . . . . . . 7 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 β‰  𝐡 β†’ Β¬ 𝐴 β‰ˆ 𝐡))
2726necon4ad 2959 . . . . . 6 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (𝐴 β‰ˆ 𝐡 β†’ ran 𝐹 = 𝐡))
285, 27mpd 15 . . . . 5 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ ran 𝐹 = 𝐡)
29 df-fo 6546 . . . . 5 (𝐹:𝐴–onto→𝐡 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐡))
304, 28, 29sylanbrc 583 . . . 4 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴–onto→𝐡)
31 df-f1o 6547 . . . 4 (𝐹:𝐴–1-1-onto→𝐡 ↔ (𝐹:𝐴–1-1→𝐡 ∧ 𝐹:𝐴–onto→𝐡))
321, 30, 31sylanbrc 583 . . 3 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴–1-1-onto→𝐡)
3332ex 413 . 2 ((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) β†’ (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-onto→𝐡))
34 f1of1 6829 . 2 (𝐹:𝐴–1-1-onto→𝐡 β†’ 𝐹:𝐴–1-1→𝐡)
3533, 34impbid1 224 1 ((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) β†’ (𝐹:𝐴–1-1→𝐡 ↔ 𝐹:𝐴–1-1-onto→𝐡))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   βŠ† wss 3947   ⊊ wpss 3948   class class class wbr 5147  ran crn 5676   Fn wfn 6535  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“ontoβ†’wfo 6538  β€“1-1-ontoβ†’wf1o 6539   β‰ˆ cen 8932   β‰Ό cdom 8933   β‰Ί csdm 8934  Fincfn 8935
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939
This theorem is referenced by:  hashfac  14415  crth  16707  eulerthlem2  16711  fidomndrnglem  20917  mdetunilem8  22112  basellem4  26577  lgsqrlem4  26841  lgseisenlem2  26868
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