mbfmcnvima - Mathbox for Thierry Arnoux

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Theorem mbfmcnvima 34552

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 6045	. . 3 ⊢ (𝑥 = 𝐴 → (^◡𝐹 “ 𝑥) = (^◡𝐹 “ 𝐴))
2	1	eleq1d 2847	. 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 34548	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)))
7	3, 6	mpbid 234	. . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆))
8	7	simprd 499	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑇 (^◡𝐹 “ 𝑥) ∈ 𝑆)
9		mbfmcnvima.4	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑇)
10	2, 8, 9	rspcdva 3582	1 ⊢ (𝜑 → (^◡𝐹 “ 𝐴) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ∀wral 3076 ∪ cuni 4865 ^◡ccnv 5646 ran crn 5648 “ cima 5650 (class class class)co 7396 ↑_m cmap 8808 sigAlgebracsiga 34405 MblFnMcmbfm 34546

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pr 5390

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-sbc 3745 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fv 6529 df-ov 7399 df-oprab 7400 df-mpo 7401 df-mbfm 34547

This theorem is referenced by: imambfm 34559 mbfmco 34561 mbfmco2 34562 sxbrsiga 34587 sibfinima 34636 sibfof 34637 orvcoel 34759 orvccel 34760

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