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Theorem f1finf1o 9222
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 5325. (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 486 . . . 4 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴–1-1→𝐡)
2 f1f 6743 . . . . . . 7 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴⟢𝐡)
32adantl 483 . . . . . 6 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴⟢𝐡)
43ffnd 6674 . . . . 5 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹 Fn 𝐴)
5 simpll 766 . . . . . 6 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐴 β‰ˆ 𝐡)
63frnd 6681 . . . . . . . . 9 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ ran 𝐹 βŠ† 𝐡)
7 df-pss 3934 . . . . . . . . . 10 (ran 𝐹 ⊊ 𝐡 ↔ (ran 𝐹 βŠ† 𝐡 ∧ ran 𝐹 β‰  𝐡))
87baib 537 . . . . . . . . 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 9163 . . . . . . . . . . . 12 ((𝐡 ∈ Fin ∧ ran 𝐹 ⊊ 𝐡) β†’ ran 𝐹 β‰Ί 𝐡)
1110ex 414 . . . . . . . . . . 11 (𝐡 ∈ Fin β†’ (ran 𝐹 ⊊ 𝐡 β†’ ran 𝐹 β‰Ί 𝐡))
1211ad2antlr 726 . . . . . . . . . 10 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 ⊊ 𝐡 β†’ ran 𝐹 β‰Ί 𝐡))
13 enfii 9140 . . . . . . . . . . . 12 ((𝐡 ∈ Fin ∧ 𝐴 β‰ˆ 𝐡) β†’ 𝐴 ∈ Fin)
1413ancoms 460 . . . . . . . . . . 11 ((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) β†’ 𝐴 ∈ Fin)
15 f1f1orn 6800 . . . . . . . . . . . 12 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
16 f1oenfi 9133 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹) β†’ 𝐴 β‰ˆ ran 𝐹)
1714, 15, 16syl2an 597 . . . . . . . . . . 11 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐴 β‰ˆ ran 𝐹)
18 endom 8926 . . . . . . . . . . . . 13 (𝐴 β‰ˆ ran 𝐹 β†’ 𝐴 β‰Ό ran 𝐹)
19 domsdomtrfi 9156 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝐴 β‰Ό ran 𝐹 ∧ ran 𝐹 β‰Ί 𝐡) β†’ 𝐴 β‰Ί 𝐡)
2018, 19syl3an2 1165 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝐴 β‰ˆ ran 𝐹 ∧ ran 𝐹 β‰Ί 𝐡) β†’ 𝐴 β‰Ί 𝐡)
21203expia 1122 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ 𝐴 β‰ˆ ran 𝐹) β†’ (ran 𝐹 β‰Ί 𝐡 β†’ 𝐴 β‰Ί 𝐡))
2214, 17, 21syl2an2r 684 . . . . . . . . . 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 8928 . . . . . . . . 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 260 . . . . . . 7 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 β‰  𝐡 β†’ Β¬ 𝐴 β‰ˆ 𝐡))
2726necon4ad 2963 . . . . . 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 6507 . . . . 5 (𝐹:𝐴–onto→𝐡 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐡))
304, 28, 29sylanbrc 584 . . . 4 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴–onto→𝐡)
31 df-f1o 6508 . . . 4 (𝐹:𝐴–1-1-onto→𝐡 ↔ (𝐹:𝐴–1-1→𝐡 ∧ 𝐹:𝐴–onto→𝐡))
321, 30, 31sylanbrc 584 . . 3 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴–1-1-onto→𝐡)
3332ex 414 . 2 ((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) β†’ (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-onto→𝐡))
34 f1of1 6788 . 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   βŠ† wss 3915   ⊊ wpss 3916   class class class wbr 5110  ran crn 5639   Fn wfn 6496  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€“ontoβ†’wfo 6499  β€“1-1-ontoβ†’wf1o 6500   β‰ˆ cen 8887   β‰Ό cdom 8888   β‰Ί csdm 8889  Fincfn 8890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894
This theorem is referenced by:  hashfac  14364  crth  16657  eulerthlem2  16661  fidomndrnglem  20793  mdetunilem8  21984  basellem4  26449  lgsqrlem4  26713  lgseisenlem2  26740
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