mbfmcnvima - Mathbox for Thierry Arnoux

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Theorem mbfmcnvima 34195

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		imaeq2 6040	. . 3 ⊢ (𝑥 = 𝐴 → (^◡𝐹 “ 𝑥) = (^◡𝐹 “ 𝐴))
2	1	eleq1d 2818	. 2 ⊢ (𝑥 = 𝐴 → ((^◡𝐹 “ 𝑥) ∈ 𝑆 ↔ (^◡𝐹 “ 𝐴) ∈ 𝑆))
3		mbfmf.3	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))
4		mbfmf.1	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
5		mbfmf.2	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
6	4, 5	ismbfm 34190	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)))
7	3, 6	mpbid 232	. . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆))
8	7	simprd 495	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)
9		mbfmcnvima.4	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑇)
10	2, 8, 9	rspcdva 3600	1 ⊢ (𝜑 → (^◡𝐹 “ 𝐴) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3050 ∪ cuni 4880 ^◡ccnv 5650 ran crn 5652 “ cima 5654 (class class class)co 7399 ↑_m cmap 8834 sigAlgebracsiga 34047 MblFnMcmbfm 34188

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5263 ax-nul 5273 ax-pr 5399

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rab 3414 df-v 3459 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4881 df-br 5117 df-opab 5179 df-id 5545 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6480 df-fun 6529 df-fv 6535 df-ov 7402 df-oprab 7403 df-mpo 7404 df-mbfm 34189

This theorem is referenced by: imambfm 34202 mbfmco 34204 mbfmco2 34205 sxbrsiga 34230 sibfinima 34279 sibfof 34280 orvcoel 34402 orvccel 34403

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