Theorem ssdomg

Description: A set dominates its subsets. Theorem 16 of Suppes p. 94. (Contributed by NM, 19-Jun-1998) (Revised by Mario Carneiro, 24-Jun-2015)

	Ref	Expression
Assertion	ssdomg	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ssexg	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 ∈ V )
2		simpr	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → 𝐵 ∈ 𝑉 )
3		f1oi	⊢ ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴
4		dff1o3	⊢ ( ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 ↔ ( ( I ↾ 𝐴 ) : 𝐴 –onto→ 𝐴 ∧ Fun ◡ ( I ↾ 𝐴 ) ) )
5	3 4	mpbi	⊢ ( ( I ↾ 𝐴 ) : 𝐴 –onto→ 𝐴 ∧ Fun ◡ ( I ↾ 𝐴 ) )
6	5	simpli	⊢ ( I ↾ 𝐴 ) : 𝐴 –onto→ 𝐴
7		fof	⊢ ( ( I ↾ 𝐴 ) : 𝐴 –onto→ 𝐴 → ( I ↾ 𝐴 ) : 𝐴 ⟶ 𝐴 )
8	6 7	ax-mp	⊢ ( I ↾ 𝐴 ) : 𝐴 ⟶ 𝐴
9		fss	⊢ ( ( ( I ↾ 𝐴 ) : 𝐴 ⟶ 𝐴 ∧ 𝐴 ⊆ 𝐵 ) → ( I ↾ 𝐴 ) : 𝐴 ⟶ 𝐵 )
10	8 9	mpan	⊢ ( 𝐴 ⊆ 𝐵 → ( I ↾ 𝐴 ) : 𝐴 ⟶ 𝐵 )
11		funi	⊢ Fun I
12		cnvi	⊢ ◡ I = I
13	12	funeqi	⊢ ( Fun ◡ I ↔ Fun I )
14	11 13	mpbir	⊢ Fun ◡ I
15		funres11	⊢ ( Fun ◡ I → Fun ◡ ( I ↾ 𝐴 ) )
16	14 15	ax-mp	⊢ Fun ◡ ( I ↾ 𝐴 )
17		df-f1	⊢ ( ( I ↾ 𝐴 ) : 𝐴 –1-1→ 𝐵 ↔ ( ( I ↾ 𝐴 ) : 𝐴 ⟶ 𝐵 ∧ Fun ◡ ( I ↾ 𝐴 ) ) )
18	10 16 17	sylanblrc	⊢ ( 𝐴 ⊆ 𝐵 → ( I ↾ 𝐴 ) : 𝐴 –1-1→ 𝐵 )
19	18	adantr	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → ( I ↾ 𝐴 ) : 𝐴 –1-1→ 𝐵 )
20		f1dom2g	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ 𝑉 ∧ ( I ↾ 𝐴 ) : 𝐴 –1-1→ 𝐵 ) → 𝐴 ≼ 𝐵 )
21	1 2 19 20	syl3anc	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 ≼ 𝐵 )
22	21	expcom	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵 ) )