Theorem f1imaenfi 9136

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 8962). (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 6796	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
2		f1oenfi 9120	. . . . 5 ⊢ ((𝐶 ∈ Fin ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶))
3		ensymfib 9125	. . . . . 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 3911   class class class wbr 5102   ↾ cres 5633   “ cima 5634  –1-1→wf1 6496  –1-1-onto→wf1o 6498   ≈ cen 8892  Fincfn 8895

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 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-en 8896  df-fin 8899

This theorem is referenced by:  phplem2  9146