Theorem 2ndmbfm 34274

Description: The second projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)

Hypotheses

Ref	Expression
1stmbfm.1	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
1stmbfm.2	⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)

Assertion

Ref	Expression
2ndmbfm	⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))

Proof of Theorem 2ndmbfm

Dummy variables 𝑧 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f2ndres 7946	. . . 4 ⊢ (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑇
2		1stmbfm.1	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
3		1stmbfm.2	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
4		sxuni 34206	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
5	2, 3, 4	syl2anc 584	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
6	5	feq2d 6635	. . . 4 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑇 ↔ (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇))
7	1, 6	mpbii 233	. . 3 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇)
8		unielsiga 34141	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇)
9	3, 8	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
10		sxsiga 34204	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ∪ ran sigAlgebra)
11	2, 3, 10	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑆 ×s 𝑇) ∈ ∪ ran sigAlgebra)
12		unielsiga 34141	. . . . 5 ⊢ ((𝑆 ×s 𝑇) ∈ ∪ ran sigAlgebra → ∪ (𝑆 ×s 𝑇) ∈ (𝑆 ×s 𝑇))
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → ∪ (𝑆 ×s 𝑇) ∈ (𝑆 ×s 𝑇))
14	9, 13	elmapd 8764	. . 3 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑇 ↑m ∪ (𝑆 ×s 𝑇)) ↔ (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇))
15	7, 14	mpbird 257	. 2 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑇 ↑m ∪ (𝑆 ×s 𝑇)))
16		ffn 6651	. . . . . . . 8 ⊢ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑇 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) Fn (∪ 𝑆 × ∪ 𝑇))
17		elpreima 6991	. . . . . . . 8 ⊢ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)) Fn (∪ 𝑆 × ∪ 𝑇) → (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎)))
18	1, 16, 17	mp2b 10	. . . . . . 7 ⊢ (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎))
19		fvres 6841	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) = (2nd ‘𝑧))
20	19	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ (2nd ‘𝑧) ∈ 𝑎))
21		1st2nd2 7960	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
22		xp1st 7953	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (1st ‘𝑧) ∈ ∪ 𝑆)
23		elxp6 7955	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ ∪ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑎)))
24		anass 468	. . . . . . . . . . . 12 ⊢ (((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ ∪ 𝑆) ∧ (2nd ‘𝑧) ∈ 𝑎) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ ∪ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑎)))
25	23, 24	bitr4i 278	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ ∪ 𝑆) ∧ (2nd ‘𝑧) ∈ 𝑎))
26	25	baib 535	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ ∪ 𝑆) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (2nd ‘𝑧) ∈ 𝑎))
27	21, 22, 26	syl2anc 584	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (2nd ‘𝑧) ∈ 𝑎))
28	20, 27	bitr4d 282	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
29	28	pm5.32i 574	. . . . . . 7 ⊢ ((𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
30	18, 29	bitri 275	. . . . . 6 ⊢ (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
31		sgon 34137	. . . . . . . . . . 11 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → 𝑇 ∈ (sigAlgebra‘∪ 𝑇))
32		sigasspw 34129	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (sigAlgebra‘∪ 𝑇) → 𝑇 ⊆ 𝒫 ∪ 𝑇)
33		pwssb 5047	. . . . . . . . . . . 12 ⊢ (𝑇 ⊆ 𝒫 ∪ 𝑇 ↔ ∀𝑎 ∈ 𝑇 𝑎 ⊆ ∪ 𝑇)
34	33	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑇 ⊆ 𝒫 ∪ 𝑇 → ∀𝑎 ∈ 𝑇 𝑎 ⊆ ∪ 𝑇)
35	3, 31, 32, 34	4syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 𝑎 ⊆ ∪ 𝑇)
36	35	r19.21bi 3224	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ ∪ 𝑇)
37		xpss2 5634	. . . . . . . . 9 ⊢ (𝑎 ⊆ ∪ 𝑇 → (∪ 𝑆 × 𝑎) ⊆ (∪ 𝑆 × ∪ 𝑇))
38	36, 37	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (∪ 𝑆 × 𝑎) ⊆ (∪ 𝑆 × ∪ 𝑇))
39	38	sseld 3928	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) → 𝑧 ∈ (∪ 𝑆 × ∪ 𝑇)))
40	39	pm4.71rd 562	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎))))
41	30, 40	bitr4id 290	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
42	41	eqrdv 2729	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) = (∪ 𝑆 × 𝑎))
43	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra)
44	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra)
45		eqid 2731	. . . . . . . 8 ⊢ ∪ 𝑆 = ∪ 𝑆
46		issgon 34136	. . . . . . . 8 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) ↔ (𝑆 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑆 = ∪ 𝑆))
47	2, 45, 46	sylanblrc 590	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
48		baselsiga 34128	. . . . . . 7 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) → ∪ 𝑆 ∈ 𝑆)
49	47, 48	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
50	49	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∪ 𝑆 ∈ 𝑆)
51		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇)
52		elsx 34207	. . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) ∧ (∪ 𝑆 ∈ 𝑆 ∧ 𝑎 ∈ 𝑇)) → (∪ 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
53	43, 44, 50, 51, 52	syl22anc 838	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (∪ 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
54	42, 53	eqeltrd 2831	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
55	54	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
56	11, 3	ismbfm 34264	. 2 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇) ↔ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑇 ↑m ∪ (𝑆 ×s 𝑇)) ∧ ∀𝑎 ∈ 𝑇 (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))))
57	15, 55, 56	mpbir2and 713	1 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897  𝒫 cpw 4547  ⟨cop 4579  ∪ cuni 4856   × cxp 5612  ^◡ccnv 5613  ran crn 5615   ↾ cres 5616   “ cima 5617   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920   ↑_m cmap 8750  sigAlgebracsiga 34121   ×_s csx 34201  MblFnMcmbfm 34262

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-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

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-reu 3347  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 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-siga 34122  df-sigagen 34152  df-sx 34202  df-mbfm 34263

This theorem is referenced by: (None)