Theorem mbfmco 34775

Description: The composition of two measurable functions is measurable. See cnmpt11 23889. (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 34765	. . . 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 34765	. . . 4 ⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆)
8		fco 6727	. . . 4 ⊢ ((𝐺:∪ 𝑆⟶∪ 𝑇 ∧ 𝐹:∪ 𝑅⟶∪ 𝑆) → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇)
9	4, 7, 8	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇)
10		unielsiga 34638	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇)
11	2, 10	syl 18	. . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
12		unielsiga 34638	. . . . 5 ⊢ (𝑅 ∈ ∪ ran sigAlgebra → ∪ 𝑅 ∈ 𝑅)
13	5, 12	syl 18	. . . 4 ⊢ (𝜑 → ∪ 𝑅 ∈ 𝑅)
14	11, 13	elmapd 8839	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ↔ (𝐺 ∘ 𝐹):∪ 𝑅⟶∪ 𝑇))
15	9, 14	mpbird 260	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅))
16		cnvco 5869	. . . . . 6 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
17	16	imaeq1i 6053	. . . . 5 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = ((◡𝐹 ∘ ◡𝐺) “ 𝑎)
18		imaco 6247	. . . . 5 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎))
19	17, 18	eqtri 2783	. . . 4 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑎) = (◡𝐹 “ (◡𝐺 “ 𝑎))
20	5	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑅 ∈ ∪ ran sigAlgebra)
21	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra)
22	6	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆))
23	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra)
24	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐺 ∈ (𝑆MblFnM𝑇))
25		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇)
26	21, 23, 24, 25	mbfmcnvima 34766	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐺 “ 𝑎) ∈ 𝑆)
27	20, 21, 22, 26	mbfmcnvima 34766	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡𝐹 “ (◡𝐺 “ 𝑎)) ∈ 𝑅)
28	19, 27	eqeltrid 2864	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)
29	28	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)
30	5, 2	ismbfm 34762	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (∪ 𝑇 ↑m ∪ 𝑅) ∧ ∀𝑎 ∈ 𝑇 (◡(𝐺 ∘ 𝐹) “ 𝑎) ∈ 𝑅)))
31	15, 29, 30	mpbir2and 726	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝑅MblFnM𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3076  ∪ cuni 4867  ^◡ccnv 5654  ran crn 5656   “ cima 5658   ∘ ccom 5659  ⟶wf 6529  (class class class)co 7413   ↑_m cmap 8826  sigAlgebracsiga 34618  MblFnMcmbfm 34760

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-map 8828  df-siga 34619  df-mbfm 34761

This theorem is used by:  rrvadd  34963  rrvmulc  34964