Theorem f1oenfi 8836

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 8625). (Contributed by BTernaryTau, 8-Sep-2024.)

Assertion

Ref	Expression
f1oenfi	⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)

Proof of Theorem f1oenfi

Step	Hyp	Ref	Expression
1		f1ofn 6640	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
2		fnfi 8835	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)
3	1, 2	sylan 583	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)
4	3	ancoms 462	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐹 ∈ Fin)
5		f1oen3g 8622	. 2 ⊢ ((𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
6	4, 5	sylancom 591	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2112   class class class wbr 5039   Fn wfn 6353  –1-1-onto→wf1o 6357   ≈ cen 8601  Fincfn 8604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-om 7623  df-1o 8180  df-en 8605  df-fin 8608

This theorem is referenced by:  enreffi  8838  ensymfib  8839  entrfil  8840