Theorem mbfmco 34269

Description: The composition of two measurable functions is measurable. See cnmpt11 23573. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Hypotheses

Ref	Expression
mbfmco.1	⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra)
mbfmco.2	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
mbfmco.3	⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
mbfmco.4	⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆))
mbfmco.5	⊢ (𝜑 → 𝐺 ∈ (𝑆MblFnM𝑇))

Assertion

Ref	Expression
mbfmco	⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇))

Proof of Theorem mbfmco

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfmco.2	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
2		mbfmco.3	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
3		mbfmco.5	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑆MblFnM𝑇))
4	1, 2, 3	mbfmf 34259	. . . 4 ⊢ (𝜑 → 𝐺:∪ 𝑆⟶∪ 𝑇)
5		mbfmco.1	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra)
6		mbfmco.4	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆))
7	5, 1, 6	mbfmf 34259	. . . 4 ⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆)
8		fco 6670	. . . 4 ⊢ ((𝐺:∪ 𝑆⟶∪ 𝑇 ∧ 𝐹:∪ 𝑅⟶∪ 𝑆) → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇)
9	4, 7, 8	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇)
10		unielsiga 34133	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇)
11	2, 10	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
12		unielsiga 34133	. . . . 5 ⊢ (𝑅 ∈ ∪ ran sigAlgebra → ∪ 𝑅 ∈ 𝑅)
13	5, 12	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑅 ∈ 𝑅)
14	11, 13	elmapd 8759	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ↔ (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇))
15	9, 14	mpbird 257	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅))
16		cnvco 5820	. . . . . 6 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
17	16	imaeq1i 6001	. . . . 5 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = ((◡𝐹 ∘ ◡𝐺) “ 𝑎)
18		imaco 6193	. . . . 5 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎))
19	17, 18	eqtri 2754	. . . 4 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎))
20	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑅 ∈ ∪ ran sigAlgebra)
21	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra)
22	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
23	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra)
24	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐺 ∈ (𝑆MblFnM𝑇))
25		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇)
26	21, 23, 24, 25	mbfmcnvima 34260	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐺 “ 𝑎) ∈ 𝑆)
27	20, 21, 22, 26	mbfmcnvima 34260	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐹 “ (◡𝐺 “ 𝑎)) ∈ 𝑅)
28	19, 27	eqeltrid 2835	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)
29	28	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)
30	5, 2	ismbfm 34256	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ∧ ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)))
31	15, 29, 30	mpbir2and 713	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2111  ∀wral 3047  ∪ cuni 4854  ^◡ccnv 5610  ran crn 5612   “ cima 5614   ∘ ccom 5615  ⟶wf 6472  (class class class)co 7341   ↑_m cmap 8745  sigAlgebracsiga 34113  MblFnMcmbfm 34254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-map 8747  df-siga 34114  df-mbfm 34255

This theorem is referenced by:  rrvadd  34457  rrvmulc  34458