Theorem wdomd

Description: Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015)

	Ref	Expression
Hypotheses	wdomd.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
	wdomd.o	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 )
Assertion	wdomd	⊢ ( 𝜑 → 𝐴 ≼* 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		wdomd.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
2		wdomd.o	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 )
3		abrexexg	⊢ ( 𝐵 ∈ 𝑊 → { 𝑥 ∣ ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 } ∈ V )
4	1 3	syl	⊢ ( 𝜑 → { 𝑥 ∣ ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 } ∈ V )
5	2	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 ) )
6	5	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 ) )
7		ssab	⊢ ( 𝐴 ⊆ { 𝑥 ∣ ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 ) )
8	6 7	sylibr	⊢ ( 𝜑 → 𝐴 ⊆ { 𝑥 ∣ ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 } )
9	4 8	ssexd	⊢ ( 𝜑 → 𝐴 ∈ V )
10	9 1 2	wdom2d	⊢ ( 𝜑 → 𝐴 ≼* 𝐵 )