Theorem f1sng

Description: A singleton of an ordered pair is a one-to-one function. (Contributed by AV, 17-Apr-2021)

	Ref	Expression
Assertion	f1sng	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1→ 𝑊 )

Proof

Step	Hyp	Ref	Expression
1		f1osng	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1-onto→ { 𝐵 } )
2		f1of1	⊢ ( { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1-onto→ { 𝐵 } → { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1→ { 𝐵 } )
3	1 2	syl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1→ { 𝐵 } )
4		snssi	⊢ ( 𝐵 ∈ 𝑊 → { 𝐵 } ⊆ 𝑊 )
5	4	adantl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → { 𝐵 } ⊆ 𝑊 )
6		f1ss	⊢ ( ( { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1→ { 𝐵 } ∧ { 𝐵 } ⊆ 𝑊 ) → { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1→ 𝑊 )
7	3 5 6	syl2anc	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → { ⟨ 𝐴 , 𝐵 ⟩ } : { 𝐴 } –1-1→ 𝑊 )