Theorem f1imaen3g 8955

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 8953 does not need ax-rep 5223 nor ax-pow 5309.) (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 5985	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝐶) ∈ V)
2	1	3ad2ant3 1136	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ 𝑉) → (𝐹 ↾ 𝐶) ∈ V)
3		f1ores 6787	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
4	3	3adant3 1133	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ 𝑉) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
5		f1oen3g 8905	. 2 ⊢ (((𝐹 ↾ 𝐶) ∈ V ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶))
6	2, 4, 5	syl2anc 585	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ 𝑉) → 𝐶 ≈ (𝐹 “ 𝐶))

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2114  Vcvv 3439   ⊆ wss 3900   class class class wbr 5097   ↾ cres 5625   “ cima 5626  –1-1→wf1 6488  –1-1-onto→wf1o 6490   ≈ cen 8882

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-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-en 8886

This theorem is referenced by:  fiint  9229