mbfmcnvima - Mathbox for Thierry Arnoux

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Theorem mbfmcnvima 30835

Description: The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.)

**Hypotheses**
Ref	Expression
mbfmf.1	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
mbfmf.2	⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
mbfmf.3	⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))
mbfmcnvima.4	⊢ (𝜑 → 𝐴 ∈ 𝑇)

**Assertion**
Ref	Expression
mbfmcnvima	⊢ (𝜑 → (^◡𝐹 “ 𝐴) ∈ 𝑆)

Proof of Theorem mbfmcnvima Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfmcnvima.4	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑇)
2		mbfmf.3	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))
3		mbfmf.1	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
4		mbfmf.2	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
5	3, 4	ismbfm 30830	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑_𝑚 ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)))
6	2, 5	mpbid 224	. . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑_𝑚 ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆))
7	6	simprd 490	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)
8		imaeq2 5679	. . . 4 ⊢ (𝑥 = 𝐴 → (^◡𝐹 “ 𝑥) = (^◡𝐹 “ 𝐴))
9	8	eleq1d 2863	. . 3 ⊢ (𝑥 = 𝐴 → ((^◡𝐹 “ 𝑥) ∈ 𝑆 ↔ (^◡𝐹 “ 𝐴) ∈ 𝑆))
10	9	rspcv 3493	. 2 ⊢ (𝐴 ∈ 𝑇 → (∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆 → (^◡𝐹 “ 𝐴) ∈ 𝑆))
11	1, 7, 10	sylc 65	1 ⊢ (𝜑 → (^◡𝐹 “ 𝐴) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 ∪ cuni 4628 ^◡ccnv 5311 ran crn 5313 “ cima 5315 (class class class)co 6878 ↑_𝑚 cmap 8095 sigAlgebracsiga 30686 MblFnMcmbfm 30828

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-mbfm 30829

This theorem is referenced by: imambfm 30840 mbfmco 30842 mbfmco2 30843 sxbrsiga 30868 sibfinima 30917 sibfof 30918 orvcoel 31040 orvccel 31041

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