Theorem f1dom2g

Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015)

	Ref	Expression
Assertion	f1dom2g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → 𝐴 ≼ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		f1f	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )
2		fex2	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝐹 ∈ V )
3	1 2	syl3an1	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝐹 ∈ V )
4	3	3coml	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → 𝐹 ∈ V )
5		simp3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → 𝐹 : 𝐴 –1-1→ 𝐵 )
6		f1eq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝐴 –1-1→ 𝐵 ↔ 𝐹 : 𝐴 –1-1→ 𝐵 ) )
7	4 5 6	spcedv	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 )
8		brdomg	⊢ ( 𝐵 ∈ 𝑊 → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
9	8	3ad2ant2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
10	7 9	mpbird	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → 𝐴 ≼ 𝐵 )