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Theorem f1finf1o 9215
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 5320. (Revised by BTernaryTau, 4-Jan-2025.)
Assertion
Ref Expression
f1finf1o ((𝐴𝐵𝐵 ∈ Fin) → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto𝐵))

Proof of Theorem f1finf1o
StepHypRef Expression
1 simpr 485 . . . 4 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → 𝐹:𝐴1-1𝐵)
2 f1f 6738 . . . . . . 7 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
32adantl 482 . . . . . 6 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → 𝐹:𝐴𝐵)
43ffnd 6669 . . . . 5 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → 𝐹 Fn 𝐴)
5 simpll 765 . . . . . 6 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → 𝐴𝐵)
63frnd 6676 . . . . . . . . 9 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → ran 𝐹𝐵)
7 df-pss 3929 . . . . . . . . . 10 (ran 𝐹𝐵 ↔ (ran 𝐹𝐵 ∧ ran 𝐹𝐵))
87baib 536 . . . . . . . . 9 (ran 𝐹𝐵 → (ran 𝐹𝐵 ↔ ran 𝐹𝐵))
96, 8syl 17 . . . . . . . 8 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → (ran 𝐹𝐵 ↔ ran 𝐹𝐵))
10 php3 9156 . . . . . . . . . . . 12 ((𝐵 ∈ Fin ∧ ran 𝐹𝐵) → ran 𝐹𝐵)
1110ex 413 . . . . . . . . . . 11 (𝐵 ∈ Fin → (ran 𝐹𝐵 → ran 𝐹𝐵))
1211ad2antlr 725 . . . . . . . . . 10 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → (ran 𝐹𝐵 → ran 𝐹𝐵))
13 enfii 9133 . . . . . . . . . . . 12 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)
1413ancoms 459 . . . . . . . . . . 11 ((𝐴𝐵𝐵 ∈ Fin) → 𝐴 ∈ Fin)
15 f1f1orn 6795 . . . . . . . . . . . 12 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
16 f1oenfi 9126 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝐹:𝐴1-1-onto→ran 𝐹) → 𝐴 ≈ ran 𝐹)
1714, 15, 16syl2an 596 . . . . . . . . . . 11 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → 𝐴 ≈ ran 𝐹)
18 endom 8919 . . . . . . . . . . . . 13 (𝐴 ≈ ran 𝐹𝐴 ≼ ran 𝐹)
19 domsdomtrfi 9149 . . . . . . . . . . . . 13 ((𝐴 ∈ Fin ∧ 𝐴 ≼ ran 𝐹 ∧ ran 𝐹𝐵) → 𝐴𝐵)
2018, 19syl3an2 1164 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝐴 ≈ ran 𝐹 ∧ ran 𝐹𝐵) → 𝐴𝐵)
21203expia 1121 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ 𝐴 ≈ ran 𝐹) → (ran 𝐹𝐵𝐴𝐵))
2214, 17, 21syl2an2r 683 . . . . . . . . . 10 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → (ran 𝐹𝐵𝐴𝐵))
2312, 22syld 47 . . . . . . . . 9 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → (ran 𝐹𝐵𝐴𝐵))
24 sdomnen 8921 . . . . . . . . 9 (𝐴𝐵 → ¬ 𝐴𝐵)
2523, 24syl6 35 . . . . . . . 8 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → (ran 𝐹𝐵 → ¬ 𝐴𝐵))
269, 25sylbird 259 . . . . . . 7 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → (ran 𝐹𝐵 → ¬ 𝐴𝐵))
2726necon4ad 2962 . . . . . 6 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → (𝐴𝐵 → ran 𝐹 = 𝐵))
285, 27mpd 15 . . . . 5 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → ran 𝐹 = 𝐵)
29 df-fo 6502 . . . . 5 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
304, 28, 29sylanbrc 583 . . . 4 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → 𝐹:𝐴onto𝐵)
31 df-f1o 6503 . . . 4 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
321, 30, 31sylanbrc 583 . . 3 (((𝐴𝐵𝐵 ∈ Fin) ∧ 𝐹:𝐴1-1𝐵) → 𝐹:𝐴1-1-onto𝐵)
3332ex 413 . 2 ((𝐴𝐵𝐵 ∈ Fin) → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto𝐵))
34 f1of1 6783 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
3533, 34impbid1 224 1 ((𝐴𝐵𝐵 ∈ Fin) → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wss 3910  wpss 3911   class class class wbr 5105  ran crn 5634   Fn wfn 6491  wf 6492  1-1wf1 6493  ontowfo 6494  1-1-ontowf1o 6495  cen 8880  cdom 8881  csdm 8882  Fincfn 8883
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887
This theorem is referenced by:  hashfac  14357  crth  16650  eulerthlem2  16654  fidomndrnglem  20777  mdetunilem8  21968  basellem4  26433  lgsqrlem4  26697  lgseisenlem2  26724
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