Theorem fodomnum

Description: A version of fodom that doesn't require the Axiom of Choice ax-ac . (Contributed by Mario Carneiro, 28-Feb-2013) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	fodomnum	⊢ ( 𝐴 ∈ dom card → ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐵 ≼ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		fornex	⊢ ( 𝐴 ∈ dom card → ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐵 ∈ V ) )
2	1	com12	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( 𝐴 ∈ dom card → 𝐵 ∈ V ) )
3		numacn	⊢ ( 𝐵 ∈ V → ( 𝐴 ∈ dom card → 𝐴 ∈ AC 𝐵 ) )
4	2 3	syli	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( 𝐴 ∈ dom card → 𝐴 ∈ AC 𝐵 ) )
5	4	com12	⊢ ( 𝐴 ∈ dom card → ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐴 ∈ AC 𝐵 ) )
6		fodomacn	⊢ ( 𝐴 ∈ AC 𝐵 → ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐵 ≼ 𝐴 ) )
7	5 6	syli	⊢ ( 𝐴 ∈ dom card → ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐵 ≼ 𝐴 ) )