Theorem mbfmco 31522

Description: The composition of two measurable functions is measurable. ( cf. cnmpt11 22271) (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 31513	. . . 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 31513	. . . 4 ⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆)
8		fco 6531	. . . 4 ⊢ ((𝐺:∪ 𝑆⟶∪ 𝑇 ∧ 𝐹:∪ 𝑅⟶∪ 𝑆) → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇)
9	4, 7, 8	syl2anc 586	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇)
10		unielsiga 31387	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇)
11	2, 10	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
12		unielsiga 31387	. . . . 5 ⊢ (𝑅 ∈ ∪ ran sigAlgebra → ∪ 𝑅 ∈ 𝑅)
13	5, 12	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑅 ∈ 𝑅)
14	11, 13	elmapd 8420	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ↔ (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇))
15	9, 14	mpbird 259	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅))
16		cnvco 5756	. . . . . 6 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
17	16	imaeq1i 5926	. . . . 5 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = ((◡𝐹 ∘ ◡𝐺) “ 𝑎)
18		imaco 6104	. . . . 5 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎))
19	17, 18	eqtri 2844	. . . 4 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎))
20	5	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑅 ∈ ∪ ran sigAlgebra)
21	1	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra)
22	6	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
23	2	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra)
24	3	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐺 ∈ (𝑆MblFnM𝑇))
25		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇)
26	21, 23, 24, 25	mbfmcnvima 31515	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐺 “ 𝑎) ∈ 𝑆)
27	20, 21, 22, 26	mbfmcnvima 31515	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐹 “ (◡𝐺 “ 𝑎)) ∈ 𝑅)
28	19, 27	eqeltrid 2917	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)
29	28	ralrimiva 3182	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)
30	5, 2	ismbfm 31510	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ∧ ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)))
31	15, 29, 30	mpbir2and 711	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇))

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  ∀wral 3138  ∪ cuni 4838  ^◡ccnv 5554  ran crn 5556   “ cima 5558   ∘ ccom 5559  ⟶wf 6351  (class class class)co 7156   ↑_m cmap 8406  sigAlgebracsiga 31367  MblFnMcmbfm 31508

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-map 8408  df-siga 31368  df-mbfm 31509

This theorem is referenced by:  rrvadd  31710  rrvmulc  31711