Metamath Proof Explorer


Theorem suppun

Description: The support of a class/function is a subset of the support of the union of this class/function with another class/function. (Contributed by AV, 4-Jun-2019)

Ref Expression
Hypothesis suppun.g ( 𝜑𝐺𝑉 )
Assertion suppun ( 𝜑 → ( 𝐹 supp 𝑍 ) ⊆ ( ( 𝐹𝐺 ) supp 𝑍 ) )

Proof

Step Hyp Ref Expression
1 suppun.g ( 𝜑𝐺𝑉 )
2 ssun1 ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ( ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ∪ ( 𝐺 “ ( V ∖ { 𝑍 } ) ) )
3 cnvun ( 𝐹𝐺 ) = ( 𝐹 𝐺 )
4 3 imaeq1i ( ( 𝐹𝐺 ) “ ( V ∖ { 𝑍 } ) ) = ( ( 𝐹 𝐺 ) “ ( V ∖ { 𝑍 } ) )
5 imaundir ( ( 𝐹 𝐺 ) “ ( V ∖ { 𝑍 } ) ) = ( ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ∪ ( 𝐺 “ ( V ∖ { 𝑍 } ) ) )
6 4 5 eqtri ( ( 𝐹𝐺 ) “ ( V ∖ { 𝑍 } ) ) = ( ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ∪ ( 𝐺 “ ( V ∖ { 𝑍 } ) ) )
7 2 6 sseqtrri ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ( ( 𝐹𝐺 ) “ ( V ∖ { 𝑍 } ) )
8 7 a1i ( ( ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) ∧ 𝜑 ) → ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ( ( 𝐹𝐺 ) “ ( V ∖ { 𝑍 } ) ) )
9 suppimacnv ( ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) → ( 𝐹 supp 𝑍 ) = ( 𝐹 “ ( V ∖ { 𝑍 } ) ) )
10 9 adantr ( ( ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) ∧ 𝜑 ) → ( 𝐹 supp 𝑍 ) = ( 𝐹 “ ( V ∖ { 𝑍 } ) ) )
11 unexg ( ( 𝐹 ∈ V ∧ 𝐺𝑉 ) → ( 𝐹𝐺 ) ∈ V )
12 11 adantlr ( ( ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) ∧ 𝐺𝑉 ) → ( 𝐹𝐺 ) ∈ V )
13 1 12 sylan2 ( ( ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) ∧ 𝜑 ) → ( 𝐹𝐺 ) ∈ V )
14 simplr ( ( ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) ∧ 𝜑 ) → 𝑍 ∈ V )
15 suppimacnv ( ( ( 𝐹𝐺 ) ∈ V ∧ 𝑍 ∈ V ) → ( ( 𝐹𝐺 ) supp 𝑍 ) = ( ( 𝐹𝐺 ) “ ( V ∖ { 𝑍 } ) ) )
16 13 14 15 syl2anc ( ( ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) ∧ 𝜑 ) → ( ( 𝐹𝐺 ) supp 𝑍 ) = ( ( 𝐹𝐺 ) “ ( V ∖ { 𝑍 } ) ) )
17 8 10 16 3sstr4d ( ( ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) ∧ 𝜑 ) → ( 𝐹 supp 𝑍 ) ⊆ ( ( 𝐹𝐺 ) supp 𝑍 ) )
18 17 ex ( ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) → ( 𝜑 → ( 𝐹 supp 𝑍 ) ⊆ ( ( 𝐹𝐺 ) supp 𝑍 ) ) )
19 supp0prc ( ¬ ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) → ( 𝐹 supp 𝑍 ) = ∅ )
20 0ss ∅ ⊆ ( ( 𝐹𝐺 ) supp 𝑍 )
21 19 20 eqsstrdi ( ¬ ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) → ( 𝐹 supp 𝑍 ) ⊆ ( ( 𝐹𝐺 ) supp 𝑍 ) )
22 21 a1d ( ¬ ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) → ( 𝜑 → ( 𝐹 supp 𝑍 ) ⊆ ( ( 𝐹𝐺 ) supp 𝑍 ) ) )
23 18 22 pm2.61i ( 𝜑 → ( 𝐹 supp 𝑍 ) ⊆ ( ( 𝐹𝐺 ) supp 𝑍 ) )