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 = ∪ 𝑖 ∈ 𝐼 ( 𝐹 ‘ 𝑖 ) | |
Assertion | fviunfun | ⊢ ( ( Fun 𝑈 ∧ 𝐽 ∈ 𝐼 ∧ 𝑋 ∈ dom ( 𝐹 ‘ 𝐽 ) ) → ( 𝑈 ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝐽 ) ‘ 𝑋 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fviunfun.u | ⊢ 𝑈 = ∪ 𝑖 ∈ 𝐼 ( 𝐹 ‘ 𝑖 ) | |
2 | fveq2 | ⊢ ( 𝑖 = 𝐽 → ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝐽 ) ) | |
3 | 2 | ssiun2s | ⊢ ( 𝐽 ∈ 𝐼 → ( 𝐹 ‘ 𝐽 ) ⊆ ∪ 𝑖 ∈ 𝐼 ( 𝐹 ‘ 𝑖 ) ) |
4 | 3 1 | sseqtrrdi | ⊢ ( 𝐽 ∈ 𝐼 → ( 𝐹 ‘ 𝐽 ) ⊆ 𝑈 ) |
5 | funssfv | ⊢ ( ( Fun 𝑈 ∧ ( 𝐹 ‘ 𝐽 ) ⊆ 𝑈 ∧ 𝑋 ∈ dom ( 𝐹 ‘ 𝐽 ) ) → ( 𝑈 ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝐽 ) ‘ 𝑋 ) ) | |
6 | 4 5 | syl3an2 | ⊢ ( ( Fun 𝑈 ∧ 𝐽 ∈ 𝐼 ∧ 𝑋 ∈ dom ( 𝐹 ‘ 𝐽 ) ) → ( 𝑈 ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝐽 ) ‘ 𝑋 ) ) |