Theorem f1imaen3g 9035

Description: If a set function is one-to-one, then a subset of its domain is equinumerous to the image of that subset. (This version of f1imaeng 9033 does not need ax-rep 5254 nor ax-pow 5340.) (Contributed by BTernaryTau, 13-Jan-2025.)

Assertion

Ref	Expression
f1imaen3g	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ 𝑉) → 𝐶 ≈ (𝐹 “ 𝐶))

Proof of Theorem f1imaen3g

Step	Hyp	Ref	Expression
1		resexg 6019	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝐶) ∈ V)
2	1	3ad2ant3 1135	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ 𝑉) → (𝐹 ↾ 𝐶) ∈ V)
3		f1ores 6837	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
4	3	3adant3 1132	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ 𝑉) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
5		f1oen3g 8986	. 2 ⊢ (((𝐹 ↾ 𝐶) ∈ V ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶))
6	2, 4, 5	syl2anc 584	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ 𝑉) → 𝐶 ≈ (𝐹 “ 𝐶))

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2109  Vcvv 3464   ⊆ wss 3931   class class class wbr 5124   ↾ cres 5661   “ cima 5662  –1-1→wf1 6533  –1-1-onto→wf1o 6535   ≈ cen 8961

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-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-en 8965

This theorem is referenced by:  fiint  9343