Theorem f1imaenfi 9133

Description: If a function is one-to-one, then the image of a finite subset of its domain under it is equinumerous to the subset. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1imaeng 8965). (Contributed by BTernaryTau, 29-Sep-2024.)

Assertion

Ref	Expression
f1imaenfi	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ Fin) → (𝐹 “ 𝐶) ≈ 𝐶)

Proof of Theorem f1imaenfi

Step	Hyp	Ref	Expression
1		f1ores 6798	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
2		f1oenfi 9117	. . . . 5 ⊢ ((𝐶 ∈ Fin ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶))
3		ensymfib 9122	. . . . . 6 ⊢ (𝐶 ∈ Fin → (𝐶 ≈ (𝐹 “ 𝐶) ↔ (𝐹 “ 𝐶) ≈ 𝐶))
4	3	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ Fin ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → (𝐶 ≈ (𝐹 “ 𝐶) ↔ (𝐹 “ 𝐶) ≈ 𝐶))
5	2, 4	mpbid 232	. . . 4 ⊢ ((𝐶 ∈ Fin ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → (𝐹 “ 𝐶) ≈ 𝐶)
6	1, 5	sylan2 594	. . 3 ⊢ ((𝐶 ∈ Fin ∧ (𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴)) → (𝐹 “ 𝐶) ≈ 𝐶)
7	6	3impb 1115	. 2 ⊢ ((𝐶 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ 𝐶) ≈ 𝐶)
8	7	3coml 1128	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ Fin) → (𝐹 “ 𝐶) ≈ 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114   ⊆ wss 3903   class class class wbr 5100   ↾ cres 5636   “ cima 5637  –1-1→wf1 6499  –1-1-onto→wf1o 6501   ≈ cen 8894  Fincfn 8897

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-om 7821  df-1o 8409  df-en 8898  df-fin 8901

This theorem is referenced by:  phplem2  9143