Theorem f1dom4g

Description: The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024)

	Ref	Expression
Assertion	f1dom4g	⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → 𝐴 ≼ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		f1eq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝐴 –1-1→ 𝐵 ↔ 𝐹 : 𝐴 –1-1→ 𝐵 ) )
2	1	spcegv	⊢ ( 𝐹 ∈ 𝑉 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
3	2	imp	⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 )
4	3	3ad2antl1	⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 )
5		brdom2g	⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
6	5	3adant1	⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
7	6	adantr	⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
8	4 7	mpbird	⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → 𝐴 ≼ 𝐵 )