Theorem f1cofveqaeq

Description: If the values of a composition of one-to-one functions for two arguments are equal, the arguments themselves must be equal. (Contributed by AV, 3-Feb-2021)

	Ref	Expression
Assertion	f1cofveqaeq	⊢ ( ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) → 𝑋 = 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) → 𝐹 : 𝐵 –1-1→ 𝐶 )
2		f1f	⊢ ( 𝐺 : 𝐴 –1-1→ 𝐵 → 𝐺 : 𝐴 ⟶ 𝐵 )
3		ffvelrn	⊢ ( ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ 𝑋 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑋 ) ∈ 𝐵 )
4	3	ex	⊢ ( 𝐺 : 𝐴 ⟶ 𝐵 → ( 𝑋 ∈ 𝐴 → ( 𝐺 ‘ 𝑋 ) ∈ 𝐵 ) )
5		ffvelrn	⊢ ( ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ 𝑌 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑌 ) ∈ 𝐵 )
6	5	ex	⊢ ( 𝐺 : 𝐴 ⟶ 𝐵 → ( 𝑌 ∈ 𝐴 → ( 𝐺 ‘ 𝑌 ) ∈ 𝐵 ) )
7	4 6	anim12d	⊢ ( 𝐺 : 𝐴 ⟶ 𝐵 → ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑌 ) ∈ 𝐵 ) ) )
8	2 7	syl	⊢ ( 𝐺 : 𝐴 –1-1→ 𝐵 → ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑌 ) ∈ 𝐵 ) ) )
9	8	adantl	⊢ ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) → ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑌 ) ∈ 𝐵 ) ) )
10	9	imp	⊢ ( ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( 𝐺 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑌 ) ∈ 𝐵 ) )
11		f1veqaeq	⊢ ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ ( ( 𝐺 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑌 ) ∈ 𝐵 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) → ( 𝐺 ‘ 𝑋 ) = ( 𝐺 ‘ 𝑌 ) ) )
12	1 10 11	syl2an2r	⊢ ( ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) → ( 𝐺 ‘ 𝑋 ) = ( 𝐺 ‘ 𝑌 ) ) )
13		f1veqaeq	⊢ ( ( 𝐺 : 𝐴 –1-1→ 𝐵 ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( 𝐺 ‘ 𝑋 ) = ( 𝐺 ‘ 𝑌 ) → 𝑋 = 𝑌 ) )
14	13	adantll	⊢ ( ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( 𝐺 ‘ 𝑋 ) = ( 𝐺 ‘ 𝑌 ) → 𝑋 = 𝑌 ) )
15	12 14	syld	⊢ ( ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) → 𝑋 = 𝑌 ) )