Theorem numwdom

Description: A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 27-Aug-2015)

	Ref	Expression
Assertion	numwdom	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ≼* 𝐴 ) → 𝐵 ∈ dom card )

Proof

Step	Hyp	Ref	Expression
1		brwdomi	⊢ ( 𝐵 ≼* 𝐴 → ( 𝐵 = ∅ ∨ ∃ 𝑓 𝑓 : 𝐴 –onto→ 𝐵 ) )
2		simpr	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 = ∅ ) → 𝐵 = ∅ )
3		0fi	⊢ ∅ ∈ Fin
4		finnum	⊢ ( ∅ ∈ Fin → ∅ ∈ dom card )
5	3 4	ax-mp	⊢ ∅ ∈ dom card
6	2 5	eqeltrdi	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 = ∅ ) → 𝐵 ∈ dom card )
7		fonum	⊢ ( ( 𝐴 ∈ dom card ∧ 𝑓 : 𝐴 –onto→ 𝐵 ) → 𝐵 ∈ dom card )
8	7	ex	⊢ ( 𝐴 ∈ dom card → ( 𝑓 : 𝐴 –onto→ 𝐵 → 𝐵 ∈ dom card ) )
9	8	exlimdv	⊢ ( 𝐴 ∈ dom card → ( ∃ 𝑓 𝑓 : 𝐴 –onto→ 𝐵 → 𝐵 ∈ dom card ) )
10	9	imp	⊢ ( ( 𝐴 ∈ dom card ∧ ∃ 𝑓 𝑓 : 𝐴 –onto→ 𝐵 ) → 𝐵 ∈ dom card )
11	6 10	jaodan	⊢ ( ( 𝐴 ∈ dom card ∧ ( 𝐵 = ∅ ∨ ∃ 𝑓 𝑓 : 𝐴 –onto→ 𝐵 ) ) → 𝐵 ∈ dom card )
12	1 11	sylan2	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ≼* 𝐴 ) → 𝐵 ∈ dom card )