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Metamath Proof Explorer

Theorem symgextfv

Description: The function value of the extension of a permutation, fixing the additional element, for elements in the original domain. (Contributed by AV, 6-Jan-2019)

Ref Expression
Hypotheses symgext.s ⊢ 𝑆 = ( Base ‘ ( SymGrp ‘ ( 𝑁 ∖ { 𝐾 } ) ) )
symgext.e ⊢ 𝐸 = ( 𝑥 ∈ 𝑁 ↦ if ( 𝑥 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑥 ) ) )
Assertion symgextfv ( ( 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) → ( 𝑋 ∈ ( 𝑁 ∖ { 𝐾 } ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑍 ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 symgext.s ⊢ 𝑆 = ( Base ‘ ( SymGrp ‘ ( 𝑁 ∖ { 𝐾 } ) ) )
2 symgext.e ⊢ 𝐸 = ( 𝑥 ∈ 𝑁 ↦ if ( 𝑥 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑥 ) ) )
3 eldifi ⊢ ( 𝑋 ∈ ( 𝑁 ∖ { 𝐾 } ) → 𝑋 ∈ 𝑁 )
4 fvexd ⊢ ( ( 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) → ( 𝑍 ‘ 𝑋 ) ∈ V )
5 ifexg ⊢ ( ( 𝐾 ∈ 𝑁 ∧ ( 𝑍 ‘ 𝑋 ) ∈ V ) → if ( 𝑋 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑋 ) ) ∈ V )
6 4 5 syldan ⊢ ( ( 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) → if ( 𝑋 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑋 ) ) ∈ V )
7 eqeq1 ⊢ ( 𝑥 = 𝑋 → ( 𝑥 = 𝐾 ↔ 𝑋 = 𝐾 ) )
8 fveq2 ⊢ ( 𝑥 = 𝑋 → ( 𝑍 ‘ 𝑥 ) = ( 𝑍 ‘ 𝑋 ) )
9 7 8 ifbieq2d ⊢ ( 𝑥 = 𝑋 → if ( 𝑥 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑥 ) ) = if ( 𝑋 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑋 ) ) )
10 9 2 fvmptg ⊢ ( ( 𝑋 ∈ 𝑁 ∧ if ( 𝑋 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑋 ) ) ∈ V ) → ( 𝐸 ‘ 𝑋 ) = if ( 𝑋 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑋 ) ) )
11 3 6 10 syl2anr ⊢ ( ( ( 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) ∧ 𝑋 ∈ ( 𝑁 ∖ { 𝐾 } ) ) → ( 𝐸 ‘ 𝑋 ) = if ( 𝑋 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑋 ) ) )
12 eldifsnneq ⊢ ( 𝑋 ∈ ( 𝑁 ∖ { 𝐾 } ) → ¬ 𝑋 = 𝐾 )
13 12 adantl ⊢ ( ( ( 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) ∧ 𝑋 ∈ ( 𝑁 ∖ { 𝐾 } ) ) → ¬ 𝑋 = 𝐾 )
14 13 iffalsed ⊢ ( ( ( 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) ∧ 𝑋 ∈ ( 𝑁 ∖ { 𝐾 } ) ) → if ( 𝑋 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑋 ) ) = ( 𝑍 ‘ 𝑋 ) )
15 11 14 eqtrd ⊢ ( ( ( 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) ∧ 𝑋 ∈ ( 𝑁 ∖ { 𝐾 } ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑍 ‘ 𝑋 ) )
16 15 ex ⊢ ( ( 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) → ( 𝑋 ∈ ( 𝑁 ∖ { 𝐾 } ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑍 ‘ 𝑋 ) ) )