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Theorem f1oenfi 9219
Description: If the domain of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1oeng 9011). (Contributed by BTernaryTau, 8-Sep-2024.)
Assertion
Ref Expression
f1oenfi ((𝐴 ∈ Fin ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)

Proof of Theorem f1oenfi
StepHypRef Expression
1 f1ofn 6849 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐴)
2 fnfi 9218 . . . 4 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)
31, 2sylan 580 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐴 ∈ Fin) → 𝐹 ∈ Fin)
43ancoms 458 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐹 ∈ Fin)
5 f1oen3g 9007 . 2 ((𝐹 ∈ Fin ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
64, 5sylancom 588 1 ((𝐴 ∈ Fin ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108   class class class wbr 5143   Fn wfn 6556  1-1-ontowf1o 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-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-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-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-en 8986  df-fin 8989
This theorem is referenced by:  enreffi  9223  ensymfib  9224  entrfil  9225  f1imaenfi  9235  f1finf1o  9305  sticksstones18  42165  sticksstones19  42166
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