Theorem f1imaen3g 8941

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 8939 does not need ax-rep 5218 nor ax-pow 5304.) (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 5978	. . 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 6778	. . 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 8892	. 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 3436   ⊆ wss 3903   class class class wbr 5092   ↾ cres 5621   “ cima 5622  –1-1→wf1 6479  –1-1-onto→wf1o 6481   ≈ cen 8869

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 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-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  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-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-en 8873

This theorem is referenced by:  fiint  9216