Theorem symgfvne

Description: The function values of a permutation for different arguments are different. (Contributed by AV, 8-Jan-2019)

	Ref	Expression
Hypotheses	symgbas.1	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
	symgbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	symgfvne	⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑋 ) = 𝑍 → ( 𝑌 ≠ 𝑋 → ( 𝐹 ‘ 𝑌 ) ≠ 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		symgbas.1	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
2		symgbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
3	1 2	symgbasf1o	⊢ ( 𝐹 ∈ 𝐵 → 𝐹 : 𝐴 –1-1-onto→ 𝐴 )
4		f1of1	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 → 𝐹 : 𝐴 –1-1→ 𝐴 )
5		eqeq2	⊢ ( 𝑍 = ( 𝐹 ‘ 𝑋 ) → ( ( 𝐹 ‘ 𝑌 ) = 𝑍 ↔ ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑋 ) ) )
6	5	eqcoms	⊢ ( ( 𝐹 ‘ 𝑋 ) = 𝑍 → ( ( 𝐹 ‘ 𝑌 ) = 𝑍 ↔ ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑋 ) ) )
7	6	adantl	⊢ ( ( ( 𝐹 : 𝐴 –1-1→ 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑋 ) = 𝑍 ) → ( ( 𝐹 ‘ 𝑌 ) = 𝑍 ↔ ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑋 ) ) )
8		simp1	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝐹 : 𝐴 –1-1→ 𝐴 )
9		simp3	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑌 ∈ 𝐴 )
10		simp2	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑋 ∈ 𝐴 )
11		f1veqaeq	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐴 ∧ ( 𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑋 ) → 𝑌 = 𝑋 ) )
12	8 9 10 11	syl12anc	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑋 ) → 𝑌 = 𝑋 ) )
13	12	adantr	⊢ ( ( ( 𝐹 : 𝐴 –1-1→ 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑋 ) = 𝑍 ) → ( ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑋 ) → 𝑌 = 𝑋 ) )
14	7 13	sylbid	⊢ ( ( ( 𝐹 : 𝐴 –1-1→ 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑋 ) = 𝑍 ) → ( ( 𝐹 ‘ 𝑌 ) = 𝑍 → 𝑌 = 𝑋 ) )
15	14	necon3d	⊢ ( ( ( 𝐹 : 𝐴 –1-1→ 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑋 ) = 𝑍 ) → ( 𝑌 ≠ 𝑋 → ( 𝐹 ‘ 𝑌 ) ≠ 𝑍 ) )
16	15	3exp1	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐴 → ( 𝑋 ∈ 𝐴 → ( 𝑌 ∈ 𝐴 → ( ( 𝐹 ‘ 𝑋 ) = 𝑍 → ( 𝑌 ≠ 𝑋 → ( 𝐹 ‘ 𝑌 ) ≠ 𝑍 ) ) ) ) )
17	3 4 16	3syl	⊢ ( 𝐹 ∈ 𝐵 → ( 𝑋 ∈ 𝐴 → ( 𝑌 ∈ 𝐴 → ( ( 𝐹 ‘ 𝑋 ) = 𝑍 → ( 𝑌 ≠ 𝑋 → ( 𝐹 ‘ 𝑌 ) ≠ 𝑍 ) ) ) ) )
18	17	3imp	⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑋 ) = 𝑍 → ( 𝑌 ≠ 𝑋 → ( 𝐹 ‘ 𝑌 ) ≠ 𝑍 ) ) )