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Metamath Proof Explorer

Theorem i1fima

Description: Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014)

Ref Expression
Assertion i1fima ( 𝐹 ∈ dom ∫1 → ( ◡ 𝐹 “ 𝐴 ) ∈ dom vol )

Proof

Step Hyp Ref Expression
1 i1ff ⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 : ℝ ⟶ ℝ )
2 ffun ⊢ ( 𝐹 : ℝ ⟶ ℝ → Fun 𝐹 )
3 inpreima ⊢ ( Fun 𝐹 → ( ◡ 𝐹 “ ( 𝐴 ∩ ran 𝐹 ) ) = ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ ran 𝐹 ) ) )
4 iunid ⊢ ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) { 𝑦 } = ( 𝐴 ∩ ran 𝐹 )
5 4 imaeq2i ⊢ ( ◡ 𝐹 “ ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) { 𝑦 } ) = ( ◡ 𝐹 “ ( 𝐴 ∩ ran 𝐹 ) )
6 imaiun ⊢ ( ◡ 𝐹 “ ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) { 𝑦 } ) = ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } )
7 5 6 eqtr3i ⊢ ( ◡ 𝐹 “ ( 𝐴 ∩ ran 𝐹 ) ) = ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } )
8 cnvimass ⊢ ( ◡ 𝐹 “ 𝐴 ) ⊆ dom 𝐹
9 cnvimarndm ⊢ ( ◡ 𝐹 “ ran 𝐹 ) = dom 𝐹
10 8 9 sseqtrri ⊢ ( ◡ 𝐹 “ 𝐴 ) ⊆ ( ◡ 𝐹 “ ran 𝐹 )
11 dfss2 ⊢ ( ( ◡ 𝐹 “ 𝐴 ) ⊆ ( ◡ 𝐹 “ ran 𝐹 ) ↔ ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ ran 𝐹 ) ) = ( ◡ 𝐹 “ 𝐴 ) )
12 10 11 mpbi ⊢ ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ ran 𝐹 ) ) = ( ◡ 𝐹 “ 𝐴 )
13 3 7 12 3eqtr3g ⊢ ( Fun 𝐹 → ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ 𝐴 ) )
14 1 2 13 3syl ⊢ ( 𝐹 ∈ dom ∫1 → ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ 𝐴 ) )
15 i1frn ⊢ ( 𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin )
16 inss2 ⊢ ( 𝐴 ∩ ran 𝐹 ) ⊆ ran 𝐹
17 ssfi ⊢ ( ( ran 𝐹 ∈ Fin ∧ ( 𝐴 ∩ ran 𝐹 ) ⊆ ran 𝐹 ) → ( 𝐴 ∩ ran 𝐹 ) ∈ Fin )
18 15 16 17 sylancl ⊢ ( 𝐹 ∈ dom ∫1 → ( 𝐴 ∩ ran 𝐹 ) ∈ Fin )
19 i1fmbf ⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn )
20 19 adantr ⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ) → 𝐹 ∈ MblFn )
21 1 adantr ⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ) → 𝐹 : ℝ ⟶ ℝ )
22 1 frnd ⊢ ( 𝐹 ∈ dom ∫1 → ran 𝐹 ⊆ ℝ )
23 16 22 sstrid ⊢ ( 𝐹 ∈ dom ∫1 → ( 𝐴 ∩ ran 𝐹 ) ⊆ ℝ )
24 23 sselda ⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ) → 𝑦 ∈ ℝ )
25 mbfimasn ⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : ℝ ⟶ ℝ ∧ 𝑦 ∈ ℝ ) → ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol )
26 20 21 24 25 syl3anc ⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ) → ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol )
27 26 ralrimiva ⊢ ( 𝐹 ∈ dom ∫1 → ∀ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol )
28 finiunmbl ⊢ ( ( ( 𝐴 ∩ ran 𝐹 ) ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol ) → ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol )
29 18 27 28 syl2anc ⊢ ( 𝐹 ∈ dom ∫1 → ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol )
30 14 29 eqeltrrd ⊢ ( 𝐹 ∈ dom ∫1 → ( ◡ 𝐹 “ 𝐴 ) ∈ dom vol )