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 )