Metamath Proof Explorer


Theorem afv2co2

Description: Value of a function composition, analogous to fvco2 . (Contributed by AV, 8-Sep-2022)

Ref Expression
Assertion afv2co2 ( ( 𝐺 defAt 𝑋𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( ( 𝐹𝐺 ) '''' 𝑋 ) = ( 𝐹 '''' ( 𝐺 '''' 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 imaco ( ( 𝐹𝐺 ) “ { 𝑋 } ) = ( 𝐹 “ ( 𝐺 “ { 𝑋 } ) )
2 dfatsnafv2 ( 𝐺 defAt 𝑋 → { ( 𝐺 '''' 𝑋 ) } = ( 𝐺 “ { 𝑋 } ) )
3 2 adantr ( ( 𝐺 defAt 𝑋𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → { ( 𝐺 '''' 𝑋 ) } = ( 𝐺 “ { 𝑋 } ) )
4 3 imaeq2d ( ( 𝐺 defAt 𝑋𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) = ( 𝐹 “ ( 𝐺 “ { 𝑋 } ) ) )
5 1 4 eqtr4id ( ( 𝐺 defAt 𝑋𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( ( 𝐹𝐺 ) “ { 𝑋 } ) = ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) )
6 5 eleq2d ( ( 𝐺 defAt 𝑋𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( 𝑥 ∈ ( ( 𝐹𝐺 ) “ { 𝑋 } ) ↔ 𝑥 ∈ ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) ) )
7 6 iotabidv ( ( 𝐺 defAt 𝑋𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( ℩ 𝑥 𝑥 ∈ ( ( 𝐹𝐺 ) “ { 𝑋 } ) ) = ( ℩ 𝑥 𝑥 ∈ ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) ) )
8 dfatco ( ( 𝐺 defAt 𝑋𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( 𝐹𝐺 ) defAt 𝑋 )
9 dfafv23 ( ( 𝐹𝐺 ) defAt 𝑋 → ( ( 𝐹𝐺 ) '''' 𝑋 ) = ( ℩ 𝑥 𝑥 ∈ ( ( 𝐹𝐺 ) “ { 𝑋 } ) ) )
10 8 9 syl ( ( 𝐺 defAt 𝑋𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( ( 𝐹𝐺 ) '''' 𝑋 ) = ( ℩ 𝑥 𝑥 ∈ ( ( 𝐹𝐺 ) “ { 𝑋 } ) ) )
11 dfafv23 ( 𝐹 defAt ( 𝐺 '''' 𝑋 ) → ( 𝐹 '''' ( 𝐺 '''' 𝑋 ) ) = ( ℩ 𝑥 𝑥 ∈ ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) ) )
12 11 adantl ( ( 𝐺 defAt 𝑋𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( 𝐹 '''' ( 𝐺 '''' 𝑋 ) ) = ( ℩ 𝑥 𝑥 ∈ ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) ) )
13 7 10 12 3eqtr4d ( ( 𝐺 defAt 𝑋𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( ( 𝐹𝐺 ) '''' 𝑋 ) = ( 𝐹 '''' ( 𝐺 '''' 𝑋 ) ) )