Theorem 2ndmbfm 30323

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 7191	. . . 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 30256	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
5	2, 3, 4	syl2anc 693	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
6	5	feq2d 6031	. . . 4 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑇 ↔ (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇))
7	1, 6	mpbii 223	. . 3 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇)
8		unielsiga 30191	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇)
9	3, 8	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
10		sxsiga 30254	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ∪ ran sigAlgebra)
11	2, 3, 10	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝑆 ×s 𝑇) ∈ ∪ ran sigAlgebra)
12		unielsiga 30191	. . . . 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 7871	. . 3 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑇 ↑𝑚 ∪ (𝑆 ×s 𝑇)) ↔ (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇))
15	7, 14	mpbird 247	. 2 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑇 ↑𝑚 ∪ (𝑆 ×s 𝑇)))
16		sgon 30187	. . . . . . . . . . 11 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → 𝑇 ∈ (sigAlgebra‘∪ 𝑇))
17		sigasspw 30179	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (sigAlgebra‘∪ 𝑇) → 𝑇 ⊆ 𝒫 ∪ 𝑇)
18		pwssb 4612	. . . . . . . . . . . 12 ⊢ (𝑇 ⊆ 𝒫 ∪ 𝑇 ↔ ∀𝑎 ∈ 𝑇 𝑎 ⊆ ∪ 𝑇)
19	18	biimpi 206	. . . . . . . . . . 11 ⊢ (𝑇 ⊆ 𝒫 ∪ 𝑇 → ∀𝑎 ∈ 𝑇 𝑎 ⊆ ∪ 𝑇)
20	3, 16, 17, 19	4syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 𝑎 ⊆ ∪ 𝑇)
21	20	r19.21bi 2932	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ ∪ 𝑇)
22		xpss2 5229	. . . . . . . . 9 ⊢ (𝑎 ⊆ ∪ 𝑇 → (∪ 𝑆 × 𝑎) ⊆ (∪ 𝑆 × ∪ 𝑇))
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (∪ 𝑆 × 𝑎) ⊆ (∪ 𝑆 × ∪ 𝑇))
24	23	sseld 3602	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) → 𝑧 ∈ (∪ 𝑆 × ∪ 𝑇)))
25	24	pm4.71rd 667	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎))))
26		ffn 6045	. . . . . . . 8 ⊢ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑇 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) Fn (∪ 𝑆 × ∪ 𝑇))
27		elpreima 6337	. . . . . . . 8 ⊢ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)) Fn (∪ 𝑆 × ∪ 𝑇) → (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎)))
28	1, 26, 27	mp2b 10	. . . . . . 7 ⊢ (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎))
29		fvres 6207	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) = (2nd ‘𝑧))
30	29	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ (2nd ‘𝑧) ∈ 𝑎))
31		1st2nd2 7205	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
32		xp1st 7198	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (1st ‘𝑧) ∈ ∪ 𝑆)
33		elxp6 7200	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ ∪ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑎)))
34		anass 681	. . . . . . . . . . . 12 ⊢ (((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ ∪ 𝑆) ∧ (2nd ‘𝑧) ∈ 𝑎) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ ∪ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑎)))
35	33, 34	bitr4i 267	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ ∪ 𝑆) ∧ (2nd ‘𝑧) ∈ 𝑎))
36	35	baib 944	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ ∪ 𝑆) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (2nd ‘𝑧) ∈ 𝑎))
37	31, 32, 36	syl2anc 693	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (2nd ‘𝑧) ∈ 𝑎))
38	30, 37	bitr4d 271	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
39	38	pm5.32i 669	. . . . . . 7 ⊢ ((𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
40	28, 39	bitri 264	. . . . . 6 ⊢ (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
41	25, 40	syl6rbbr 279	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
42	41	eqrdv 2620	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) = (∪ 𝑆 × 𝑎))
43	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra)
44	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra)
45		eqid 2622	. . . . . . . 8 ⊢ ∪ 𝑆 = ∪ 𝑆
46		issgon 30186	. . . . . . . 8 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) ↔ (𝑆 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑆 = ∪ 𝑆))
47	2, 45, 46	sylanblrc 697	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
48		baselsiga 30178	. . . . . . 7 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) → ∪ 𝑆 ∈ 𝑆)
49	47, 48	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
50	49	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∪ 𝑆 ∈ 𝑆)
51		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇)
52		elsx 30257	. . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) ∧ (∪ 𝑆 ∈ 𝑆 ∧ 𝑎 ∈ 𝑇)) → (∪ 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
53	43, 44, 50, 51, 52	syl22anc 1327	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (∪ 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
54	42, 53	eqeltrd 2701	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
55	54	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
56	11, 3	ismbfm 30314	. 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 957	1 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  𝒫 cpw 4158  ⟨cop 4183  ∪ cuni 4436   × cxp 5112  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857  sigAlgebracsiga 30170   ×_s csx 30251  MblFnMcmbfm 30312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-siga 30171  df-sigagen 30202  df-sx 30252  df-mbfm 30313

This theorem is referenced by: (None)