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Metamath Proof Explorer

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 ) ) )