Theorem ssdomfi

Description: A finite set dominates its subsets, proved without using the Axiom of Power Sets (unlike ssdomg ). (Contributed by BTernaryTau, 12-Nov-2024)

	Ref	Expression
Assertion	ssdomfi	⊢ ( 𝐵 ∈ Fin → ( 𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		f1oi	⊢ ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴
2		f1of1	⊢ ( ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 → ( I ↾ 𝐴 ) : 𝐴 –1-1→ 𝐴 )
3	1 2	ax-mp	⊢ ( I ↾ 𝐴 ) : 𝐴 –1-1→ 𝐴
4		f1ss	⊢ ( ( ( I ↾ 𝐴 ) : 𝐴 –1-1→ 𝐴 ∧ 𝐴 ⊆ 𝐵 ) → ( I ↾ 𝐴 ) : 𝐴 –1-1→ 𝐵 )
5	3 4	mpan	⊢ ( 𝐴 ⊆ 𝐵 → ( I ↾ 𝐴 ) : 𝐴 –1-1→ 𝐵 )
6		f1domfi	⊢ ( ( 𝐵 ∈ Fin ∧ ( I ↾ 𝐴 ) : 𝐴 –1-1→ 𝐵 ) → 𝐴 ≼ 𝐵 )
7	5 6	sylan2	⊢ ( ( 𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ≼ 𝐵 )
8	7	ex	⊢ ( 𝐵 ∈ Fin → ( 𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵 ) )