Metamath Proof Explorer


Theorem frnsuppeqg

Description: Version of frnsuppeq avoiding ax-rep by assuming F is a set rather than its domain I . (Contributed by SN, 30-Jul-2024)

Ref Expression
Assertion frnsuppeqg ( ( 𝐹𝑉𝑍𝑊 ) → ( 𝐹 : 𝐼𝑆 → ( 𝐹 supp 𝑍 ) = ( 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) ) )

Proof

Step Hyp Ref Expression
1 suppimacnv ( ( 𝐹𝑉𝑍𝑊 ) → ( 𝐹 supp 𝑍 ) = ( 𝐹 “ ( V ∖ { 𝑍 } ) ) )
2 ffun ( 𝐹 : 𝐼𝑆 → Fun 𝐹 )
3 inpreima ( Fun 𝐹 → ( 𝐹 “ ( 𝑆 ∩ ( V ∖ { 𝑍 } ) ) ) = ( ( 𝐹𝑆 ) ∩ ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ) )
4 2 3 syl ( 𝐹 : 𝐼𝑆 → ( 𝐹 “ ( 𝑆 ∩ ( V ∖ { 𝑍 } ) ) ) = ( ( 𝐹𝑆 ) ∩ ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ) )
5 cnvimass ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ dom 𝐹
6 fdm ( 𝐹 : 𝐼𝑆 → dom 𝐹 = 𝐼 )
7 fimacnv ( 𝐹 : 𝐼𝑆 → ( 𝐹𝑆 ) = 𝐼 )
8 6 7 eqtr4d ( 𝐹 : 𝐼𝑆 → dom 𝐹 = ( 𝐹𝑆 ) )
9 5 8 sseqtrid ( 𝐹 : 𝐼𝑆 → ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ( 𝐹𝑆 ) )
10 sseqin2 ( ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ( 𝐹𝑆 ) ↔ ( ( 𝐹𝑆 ) ∩ ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ) = ( 𝐹 “ ( V ∖ { 𝑍 } ) ) )
11 9 10 sylib ( 𝐹 : 𝐼𝑆 → ( ( 𝐹𝑆 ) ∩ ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ) = ( 𝐹 “ ( V ∖ { 𝑍 } ) ) )
12 4 11 eqtrd ( 𝐹 : 𝐼𝑆 → ( 𝐹 “ ( 𝑆 ∩ ( V ∖ { 𝑍 } ) ) ) = ( 𝐹 “ ( V ∖ { 𝑍 } ) ) )
13 invdif ( 𝑆 ∩ ( V ∖ { 𝑍 } ) ) = ( 𝑆 ∖ { 𝑍 } )
14 13 imaeq2i ( 𝐹 “ ( 𝑆 ∩ ( V ∖ { 𝑍 } ) ) ) = ( 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) )
15 12 14 eqtr3di ( 𝐹 : 𝐼𝑆 → ( 𝐹 “ ( V ∖ { 𝑍 } ) ) = ( 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) )
16 1 15 sylan9eq ( ( ( 𝐹𝑉𝑍𝑊 ) ∧ 𝐹 : 𝐼𝑆 ) → ( 𝐹 supp 𝑍 ) = ( 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) )
17 16 ex ( ( 𝐹𝑉𝑍𝑊 ) → ( 𝐹 : 𝐼𝑆 → ( 𝐹 supp 𝑍 ) = ( 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) ) )