Theorem f1imaenfi 9109

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 8939). (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 6778	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
2		f1oenfi 9093	. . . . 5 ⊢ ((𝐶 ∈ Fin ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶))
3		ensymfib 9098	. . . . . 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 593	. . 3 ⊢ ((𝐶 ∈ Fin ∧ (𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴)) → (𝐹 “ 𝐶) ≈ 𝐶)
7	6	3impb 1114	. 2 ⊢ ((𝐶 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ 𝐶) ≈ 𝐶)
8	7	3coml 1127	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ Fin) → (𝐹 “ 𝐶) ≈ 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109   ⊆ wss 3903   class class class wbr 5092   ↾ cres 5621   “ cima 5622  –1-1→wf1 6479  –1-1-onto→wf1o 6481   ≈ cen 8869  Fincfn 8872

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-om 7800  df-1o 8388  df-en 8873  df-fin 8876

This theorem is referenced by:  phplem2  9119