mbfmcnvima - Mathbox for Thierry Arnoux

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Theorem mbfmcnvima 33931

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 6054	. . 3 ⊢ (𝑥 = 𝐴 → (^◡𝐹 “ 𝑥) = (^◡𝐹 “ 𝐴))
2	1	eleq1d 2810	. 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 33926	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)))
7	3, 6	mpbid 231	. . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆))
8	7	simprd 494	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)
9		mbfmcnvima.4	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑇)
10	2, 8, 9	rspcdva 3603	1 ⊢ (𝜑 → (^◡𝐹 “ 𝐴) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∪ cuni 4903 ^◡ccnv 5671 ran crn 5673 “ cima 5675 (class class class)co 7415 ↑_m cmap 8841 sigAlgebracsiga 33783 MblFnMcmbfm 33924

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 df-mbfm 33925

This theorem is referenced by: imambfm 33938 mbfmco 33940 mbfmco2 33941 sxbrsiga 33966 sibfinima 34015 sibfof 34016 orvcoel 34137 orvccel 34138

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