Metamath Proof Explorer


Theorem fviunfun

Description: The function value of an indexed union is the value of one of the indexed functions. (Contributed by AV, 4-Nov-2023)

Ref Expression
Hypothesis fviunfun.u 𝑈 = 𝑖𝐼 ( 𝐹𝑖 )
Assertion fviunfun ( ( Fun 𝑈𝐽𝐼𝑋 ∈ dom ( 𝐹𝐽 ) ) → ( 𝑈𝑋 ) = ( ( 𝐹𝐽 ) ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 fviunfun.u 𝑈 = 𝑖𝐼 ( 𝐹𝑖 )
2 fveq2 ( 𝑖 = 𝐽 → ( 𝐹𝑖 ) = ( 𝐹𝐽 ) )
3 2 ssiun2s ( 𝐽𝐼 → ( 𝐹𝐽 ) ⊆ 𝑖𝐼 ( 𝐹𝑖 ) )
4 3 1 sseqtrrdi ( 𝐽𝐼 → ( 𝐹𝐽 ) ⊆ 𝑈 )
5 funssfv ( ( Fun 𝑈 ∧ ( 𝐹𝐽 ) ⊆ 𝑈𝑋 ∈ dom ( 𝐹𝐽 ) ) → ( 𝑈𝑋 ) = ( ( 𝐹𝐽 ) ‘ 𝑋 ) )
6 4 5 syl3an2 ( ( Fun 𝑈𝐽𝐼𝑋 ∈ dom ( 𝐹𝐽 ) ) → ( 𝑈𝑋 ) = ( ( 𝐹𝐽 ) ‘ 𝑋 ) )