Theorem brdom2g

Description: Dominance relation. This variation of brdomg does not require the Axiom of Union. (Contributed by BTernaryTau, 29-Nov-2024)

	Ref	Expression
Assertion	brdom2g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		f1eq2	⊢ ( 𝑥 = 𝐴 → ( 𝑓 : 𝑥 –1-1→ 𝑦 ↔ 𝑓 : 𝐴 –1-1→ 𝑦 ) )
2	1	exbidv	⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑓 𝑓 : 𝑥 –1-1→ 𝑦 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝑦 ) )
3		f1eq3	⊢ ( 𝑦 = 𝐵 → ( 𝑓 : 𝐴 –1-1→ 𝑦 ↔ 𝑓 : 𝐴 –1-1→ 𝐵 ) )
4	3	exbidv	⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝑦 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
5		df-dom	⊢ ≼ = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥 –1-1→ 𝑦 }
6	2 4 5	brabg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )