Theorem eqfunressuc

Description: Law for equality of restriction to successors. This is primarily useful when X is an ordinal, but it does not require that. (Contributed by Scott Fenton, 6-Dec-2021)

	Ref	Expression
Assertion	eqfunressuc	⊢ ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( 𝐹 ↾ 𝑋 ) = ( 𝐺 ↾ 𝑋 ) ∧ ( 𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ ( 𝐹 ‘ 𝑋 ) = ( 𝐺 ‘ 𝑋 ) ) ) → ( 𝐹 ↾ suc 𝑋 ) = ( 𝐺 ↾ suc 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		eqfunresadj	⊢ ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( 𝐹 ↾ 𝑋 ) = ( 𝐺 ↾ 𝑋 ) ∧ ( 𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ ( 𝐹 ‘ 𝑋 ) = ( 𝐺 ‘ 𝑋 ) ) ) → ( 𝐹 ↾ ( 𝑋 ∪ { 𝑋 } ) ) = ( 𝐺 ↾ ( 𝑋 ∪ { 𝑋 } ) ) )
2		df-suc	⊢ suc 𝑋 = ( 𝑋 ∪ { 𝑋 } )
3	2	reseq2i	⊢ ( 𝐹 ↾ suc 𝑋 ) = ( 𝐹 ↾ ( 𝑋 ∪ { 𝑋 } ) )
4	2	reseq2i	⊢ ( 𝐺 ↾ suc 𝑋 ) = ( 𝐺 ↾ ( 𝑋 ∪ { 𝑋 } ) )
5	1 3 4	3eqtr4g	⊢ ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( 𝐹 ↾ 𝑋 ) = ( 𝐺 ↾ 𝑋 ) ∧ ( 𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ ( 𝐹 ‘ 𝑋 ) = ( 𝐺 ‘ 𝑋 ) ) ) → ( 𝐹 ↾ suc 𝑋 ) = ( 𝐺 ↾ suc 𝑋 ) )