mbfmcnvima - Mathbox for Thierry Arnoux

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Theorem mbfmcnvima 30319

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 30314	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑_𝑚 ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)))
6	2, 5	mpbid 222	. . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑_𝑚 ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆))
7	6	simprd 479	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)
8		imaeq2 5462	. . . 4 ⊢ (𝑥 = 𝐴 → (^◡𝐹 “ 𝑥) = (^◡𝐹 “ 𝐴))
9	8	eleq1d 2686	. . 3 ⊢ (𝑥 = 𝐴 → ((^◡𝐹 “ 𝑥) ∈ 𝑆 ↔ (^◡𝐹 “ 𝐴) ∈ 𝑆))
10	9	rspcv 3305	. 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 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∪ cuni 4436 ^◡ccnv 5113 ran crn 5115 “ cima 5117 (class class class)co 6650 ↑_𝑚 cmap 7857 sigAlgebracsiga 30170 MblFnMcmbfm 30312

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-mbfm 30313

This theorem is referenced by: imambfm 30324 mbfmco 30326 mbfmco2 30327 sxbrsiga 30352 sibfinima 30401 sibfof 30402 orvcoel 30523 orvccel 30524

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